Entropic control over nanoscale colloidal crystals

Molecular Physics An International Journal at the Interface Between Chemistry and Physics

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Entropic control over nanoscale colloidal crystals Nathan A. Mahynski To cite this article: Nathan A. Mahynski (2016): Entropic control over nanoscale colloidal crystals, Molecular Physics, DOI: 10.1080/00268976.2016.1203467 To link to this article: http://dx.doi.org/10.1080/00268976.2016.1203467

Published online: 28 Jun 2016.

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Date: 28 June 2016, At: 08:23

MOLECULAR PHYSICS,  http://dx.doi.org/./..

THERMODYNAMICS 2015

Entropic control over nanoscale colloidal crystals Nathan A. Mahynski

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Department of Chemical and Biological Engineering, Princeton University, Princeton NJ, USA

ABSTRACT

ARTICLE HISTORY

Here the collective results of a recent body of work, which reveal how polymer architecture determines the most thermodynamically stable colloidal crystal structure in binary nanoscale colloid– polymer mixtures, are reviewed. At the nanoscale, the dimensions of the colloids, polymer segments and overall polymer size begin to converge. This may be exploited to thermodynamically stabilise a single desired crystal polymorph from a suite of competitors by leveraging the size and shape of the polymer. When each polymorph has a unique void symmetry, the entropic cost of polymer confinement in each crystal becomes significantly different. Thus, when a sufficient amount of polymer partitions into the crystal phase, the system’s total free energy difference between the competing structures is significantly amplified; in some cases by up to three orders of magnitude. The focus of this discussion is primarily on selectively stabilising one of the two close-packed polymorphs over the other; however, the heuristics presented here also lend themselves to applications in other crystals. This approach to polymorph selection requires no modification of the colloids, and is entirely based on entropy. Consequently, this technique is thermodynamically complementary to many ‘bottom-up’ self-assembly approaches, which rely on energetic interactions to stabilise a single crystal structure.

Received  February  Accepted  June 

1. Introduction Non-adsorbing polymers have been used to induce the aggregation of otherwise sterically stabilised colloidal suspensions since the early twentieth century [1–3]; however, the cause of this effect was poorly understood until the concept of ‘depletion’ was first developed by Asakura and Oosawa [4,5]. They explained this phenomenon is a consequence of the binary system maximising its overall entropy, which is achieved when the colloids aggregate, since this state minimises the total volume excluded to the polymer. As colloids diffuse towards one other, the polymer is sterically excluded from existing between the approaching surfaces. This creates an ‘osmotic vacuum’ resulting in a net attractive force between the surfaces. When a sufficient amount of polymer is added to the system, this effective attraction can induce

KEYWORDS

Colloidal crystals; polymorphism; polymer; close-packed; self-assembly

fluid–fluid and fluid–crystal phase transitions, depending on the relative size of the polymer with respect to the colloids [6–8]. Often an effective one-component Hamiltonian for the binary mixture may be obtained by integrating out the polymer degrees of freedom [5,8,9]. This approach greatly simplifies the study of these mixtures and for relatively small, non-interacting polymers, generally provides an accurate description of many aspects of these binary systems, including phase behaviour and surface tension [8,10]. In a number of cases, such as for large [11–14] or nonideal polymers [15–17], quantitative and in some cases qualitative, deviations from these predictions are known. Despite such shortcomings, coarse-graining these binary mixtures into an effectively one-component colloidal system is often a reasonably accurate approach for studying

CONTACT Nathan A. Mahynski [email protected] Chemical Informatics Research Group, Chemical Sciences Division, National Institute of Standards and Technology, Gaithersburg, Maryland -, USA ©  Informa UK Limited, trading as Taylor & Francis Group

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the fluid state of these mixtures [8,10,18]. However, in some cases removing the polymer degrees of freedom can conceal important aspects of colloidal crystallisation. As I will illustrate, this is due to the partitioning of the polymer between the resulting polymer-rich (colloid-lean, fluid) phase and the colloid-rich (polymer-lean, crystal) phase. Overlooking the importance of this effect is not without precedent. Indeed, this asymmetric partitioning was neglected for many years in early theoretical investigations of fluid–fluid phase transitions that occur in these mixtures, which is now known to be critically important for an accurate description of this phase behaviour [6,7]. For crystallising colloid–polymer mixtures in which the colloids can be treated as monodisperse hard spheres, the entropic nature of depletion favours the most dense packing of the colloids possible [8,19,20]. However, there exist two crystal structures in which this optimal packing may be achieved: the face-centred cubic (FCC) and the hexagonal close-packed (HCP) polymorphs √ [21,22], both of which have a packing fraction of φ = 2π/6 ≈ 0.74. For a pure colloidal system at this density, these crystals are known to be separated by a small difference in free energy, where the FCC crystal is marginally more stable by approximately 0.001 kB T per sphere due to entropy [23–27]. Thus, when a pure hard-sphere system crystallises, both polymorphs tend to form simultaneously, resulting in a defective solid rather than a crystal with long range order; in experimental settings it can take months or years for the less stable HCP defects to anneal into the more stable FCC [28], and this process is even sensitive to terrestrial gravity [19]. Since the aforementioned coarse-graining approach removes the details of the polymer from consideration in this crystallisation scenario, defective polymorphic solids are generally expected to result from supercooled colloid–polymer suspensions. This is, indeed, often reflected by experiments [20]. However, in what follows, I will review recent work that has revealed how to obtain a single chosen closepacked polymorph from a crystallising nanoscale binary mixture of colloids and polymers at will [29–33]. This can be accomplished by tuning the polymer’s internal degrees of freedom, such as architecture and size, and so has been overlooked by previous investigations. No modification of the colloids is required, and this crystal selectivity is a consequence of entropic considerations when a polymer becomes confined inside the crystal phase. This technique hinges principally on the observation that the accessible free volume to a polymer, when partitioned into either the FCC or HCP crystal, becomes unique to each polymorph at certain length scales. This results in an amplification of the free energy difference between the polymorphs in a binary system, with an identical amount

of adsorbed polymer, by up to three orders of magnitude over a system of pure colloids [29]. In Section 2, I discuss the origin of this effect in mixtures of colloids and linear homopolymers. I expand upon this phenomenon in Section 3 and discuss how modification of the polymer’s architecture can be exploited to stabilise either polymorph at will. Section 4 addresses the potential impact that energetic interactions between the colloids and polymers have on this, otherwise purely entropic, mechanism. Finally, Section 5 summarises the conclusions of this body of work and provides an outlook for future investigations.

2. Polymers as entropic stabilisers The FCC and HCP crystals are initially formed in an identical fashion; two layers of hexagonally packed spheres stack upon each other as depicted in Figure 1. Colloids in the second layer, referred to as layer ‘B’, rest upon the intersection of three spheres in the lower layer ‘A’, eclipsing half of all trigonal gaps formed by tangent triplets of colloids in layer A. The subsequent layers can be stacked in two possible ways while remaining closepacked. On the one hand, the next two layers can be placed in the same relative orientation as layers A and B, leading to an ‘ABAB’ stacking pattern. When repeated, this forms the HCP crystal. Alternatively, a third layer ‘C’ could be rotated 60 degrees relative to layer A and placed on the top of layer B. In this configuration, layer C eclipses the remainder of the trigonal gaps visible between layers A and B in Figure 1. When this ‘ABC’ pattern is repeated, the FCC crystal is formed. Each colloid has the same number of nearest neighbours and overall packing fraction in both polymorphs. However, a subtle difference in the arrangement of the interstitial voids becomes apparent. In both crystals, these interstices may be described as one of two platonic solids: octahedrons or tetrahedrons. These may be visualised by imagining the centre of mass of each colloid as being connected to its nearest neighbours by a wire frame. The octahedral voids (OVs) are formed between layers where the trigonal gaps are visible, as indicated in Figure 1, while tetrahedral voids (TVs) are formed where a colloid from one layer rests between a triplet of tangent colloids in its neighbouring layer. Both polymorphs have an identical overall number of TVs and OVs, in a ratio of 2:1 per colloid in the crystal. However, the manner in which these voids are arranged is unique. Between a pair of stacking planes in both polymorphs, each OV is surrounded by six TVs. Owing to the ABC stacking pattern in the FCC crystal, the remaining two faces parallel to the stacking plane are also each connected to a TV. Hence, all OVs are connected to TVs and vice versa in the FCC crystal. However, the ABAB stacking

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Figure . Void distributions and stacking patterns that occur in close-packed polymorphs. (Left) In the FCC crystal, each octahedral void (OV) is surrounded by tetrahedral voids (TV) and vice versa. (Right) In the HCP crystal, the OVs and TVs each stack in columns. An adsorbed linear homopolymer (bead chain) is also depicted. (Middle) Two layers, ‘A’and ‘B’, are each hexagonally packed in the stacking plane, which is parallel to the plane of the page. Half of the trigonal gaps formed by tangent triplets of colloids in layer A are visible through layer B; OVs are centred in these interstices. The remaining gaps are eclipsed by colloids in layer B, forming TVs. A set of ABA layers, which occurs only in the HCP crystal, is shown below with a linear homopolymer adsorbed in a column of OVs.

pattern of the HCP polymorph implies that both of these OV faces are shared with other OVs. The result is that OVs stack in ‘columns’ as do the TVs (cf. Figure 1). The significance of this can be appreciated by considering the relative size of these voids. The radius of the largest sphere that can be inscribed inside a TV without interacting with √ ≈0.112, the colloids that form its vertices is rtv /σc = 3/2−1 2 where σ c is the diameter of the colloids (edge length of the polyhedron). Whereas the largest sphere which can be √ inscribed inside an OV has a radius of rov /σc = 2−1 2 ≈0.207. Thus, an OV provides approximately six times the free volume a TV does. This ratio is further amplified after accounting for the fact that monomers, or Kuhn segments, of an adsorbed polymer will also have some excluded volume with the colloids. Consequently, when a polymer is adsorbed in the crystal phase, it is expected that the polymer will preferentially adsorb in the OVs of the crystal. If the polymer is sufficiently large, a single void will be insufficient to contain the polymer, and it will be forced to spread between neighbouring pairs of voids. Since each OV in the FCC crystal is surrounded entirely by smaller TVs, whereas the HCP environment provides simultaneous access to pairs of large OVs, it is plausible that this increase in locally accessible free volume could significantly lower the free energy of a polymer confined in the HCP crystal over one confined in the FCC [29]. This hypothesis was explored in [29], which I have summarised here. Colloids and monomers were modelled as nearly hard spheres interacting through a cut and translated Lennard–Jones potential:  Ui, j (r) = 4

1 r−

12

 −

1 r−

6  +

(1)

where  = (σ i + σ j )/2 − 1, and σ i refers to the diameter of species i. This potential terminates above rcut = 21/6 +

 so as to be purely repulsive. Polymers were treated as linear, freely jointed, bead-spring chains each with M monomers, where each monomer (Kuhn segment) had a diameter σ m = 1, and was connected by finitely extensible nonlinear elastic (FENE) bonds to its neighbours:    1 2 r− 2 Ubond (r) = − kr0 ln 1 − 2 r0

(2)

The Kremer–Grest model for FENE bonds was employed, in which k = 30ε and r0 = 1.5σ m [34]. Canonical (NVT) molecular dynamics (MD) were performed on a range of systems at various colloid packing fractions, φc = ρc πσc3 /6, and polymer volume fractions, φp = 4ρp πR3g /3, where ρ c is the colloid number density, Rg is the polymer radius of gyration, and ρ p is the polymer number density. Figure 2 depicts the results of these simulations for the case where σ c /σ m = 6.45 and M = 10 at a reduced temperature of kB T/ε = 1. After allowing the simulation to run for a sufficient period of time [29], a phase boundary becomes clear. At high polymer density, the colloids crystallise into a close-packed structure, whose radial distribution functions are depicted in Figure 2(b). Intriguingly, regardless colloid or polymer density, the crystals eventually obtain nearly identical structures. Closer inspection of the relative heights and location of the peaks in g(r) reveals that, in fact, these crystals are largely HCP and not a mix of the two polymorphs as would be expected in the case of a crystallising suspension of pure colloids. This suggests that, indeed, these linear homopolymers selectively stabilise the HCP crystal over the FCC in a binary mixture. Monte Carlo (MC) simulations confirm this and explain why M = 10 was chosen. In these MC simulations, the free energy of a single polymer chain

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Figure . Results of molecular dynamics simulations for a model system of athermal linear homopolymers with M =  beads and colloids with a diameter of σ c = .. Reproduced from []. (a) For each colloid volume fraction, φ c , at sufficiently high polymer volume fraction, φ p , the system will crystallise over the course of the simulation. An approximate phase boundary is given by the black line to guide the eye. (b) Representative colloid–colloid radial distribution functions from various simulations which crystallise. The radial distribution functions from each defect-free, close-packed polymorph are shown below for comparison.

(chemical potential) was measured by inserting the polymer bead-by-bead into a periodic cell of each polymorph. Initially, locations were chosen at random in the periodic cell, and the incremental ensemble-averaged Boltzmann factor to insert a monomer at that location was used to obtain the differential change in free energy that occurs as a result of inserting that monomer into the system: μex i = −kB T lnexp (−Uins /kB T )

(3)

After the first bead had been sampled, it was inserted into the system and allowed to freely diffuse throughout the cell. At regular intervals, a new location was proposed for the subsequent monomer, within one bond length, r0 , of the first. This was repeated to measure the ‘incremental’ chemical potential to insert that bead; the process was repeated to grow the chain to a desired length. Various MC moves including cut-and-regrowth and displacements of the entire polymer’s centre of mass were also used to accelerate the relaxation of the fully inserted portion of the chain between insertion attempts (cf. [29] for full details). The total excess chemical potential of a chain, with length M, is obtained by summing the incremental contribution from each of the monomers [35]: μex =

M 

μex i

(4)

i

In Figure 3(a), the incremental and total excess chemical potential of a chain adsorbed in each close-packed polymorph is depicted. When the chain is relatively small, the two crystals provide similar environments for the polymer and there is no statistically significant difference between the two polymorphs. Thus, small chains have no impact on the relative stability of the crystals. However, at M = 10, the excess chemical potential difference has risen

to approximately 0.1 kB T, where the chain has a lower free energy in the HCP crystal. Although this difference may seem inconsequential, recall that it is already two orders of magnitude larger, on a per-adsorbed-polymer basis, than the inherent free energy difference between the polymorphs in a pure colloidal system, on a percolloid basis [24]. When the chain is small, it largely resides in a single OV; since there is one OV per colloid in both polymorphs, the chemical potential difference of an adsorbed polymer and the inherent colloid free energy difference can be fairly compared. In molecular dynamics simulations where crystallisation occurred, many chains were observed to adsorb in the crystal, each in their own OV. As the chain length continues to increase, the polymer is forced to grow out of its OV into surrounding voids (cf. Figure 3(b)). In the FCC crystal, all OVs are surrounded entirely by TVs, so the chain must enter a more confining TV. Between M ࣈ 10 and M ࣈ 15, the chain fills a neighbouring TV until reaching the next OV, indicated by the plateau in Figure 3(b) above M ࣈ 15. However, in the HCP crystal, the polymer needs not necessarily expand into a TV once it fills its initial OV. Since OVs share faces and connect directly across the stacking plane shown in Figure 1, as the polymer becomes longer it may simply enter one of its two large OV neighbours. This increase in locally accessible free volume lowers the chemical potential of the polymer by more than 4.5 kB T relative to a polymer of the same size adsorbed in the FCC crystal. At a colloid-to-monomer size ratio of σ c /σ m = 6.45, a trigonal gap between tangent triplets of colloids within a crystal is only just large enough to allow a monomer to diffuse through the gap; so it is plausible that this polymorphic bias could be highly sensitive to monomer packing effects inside the voids. In Figure 3(c), the difference in polymer excess chemical potential between the

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Figure . Relative crystal stability in the presence of linear homopolymers. Reproduced from Ref. []. (a) The incremental (top panel) and total (lower panel) excess chemical potential of a single linear homopolymer, with M beads, adsorbed in each close-packed polymorph when σ c = . (cf. main text). At M = Mc = , the difference in a single polymer’s excess chemical potential between the two environments is approximately . kB T, and at M = Mm = , the total difference goes through a maximum of roughly . kB T. For M < Mc , the excess chemical potentials are identical to within statistical error (% confidence). (b) The ensemble-averaged number of monomers found in tetrahedral voids when the chain has length M during Monte Carlo simulations to measure the excess chemical potentials reported in (a). Symbols are also the same as reported therein. (c) The difference in a polymer’s total excess chemical potential between the two ex polymorphs, μex /kB T = (μex FCC − μHCP )/kB T , as the asymmetry between the colloid and monomer diameter grows. The chain length where the maximum difference occurs, Mm , can be fit to a power law indicated by the dashed black line. When σ c = . and the colloids have a polydispersity of δ = .%, the difference between the polymorphs remains positive, though the magnitude is reduced (dashed cyan curve). (d) Fraction of the colloids that are in the FCC and HCP states during a molecular dynamics simulation initialised from a defect-free crystallite of each polymorph while in the presence of M =  linear homopolymers (φ c = ., φ p = .).

FCC and HCP crystals is shown as this size asymmetry increases. The HCP bias persists as the colloid diameter increases relative to the size of a monomer. Although the maximum difference decays, the chain length it occurs at can be fit to a power law of the following form, indicated by the dashed black line in Figure 3(c):  −1/2 Mm ∼ μex /kB T − 2

(5)

This suggests the difference plateaus to roughly 2 kB T and illustrates that, although the polymer’s free energy difference between the polymorphs is amplified by monomerlevel packing, it is not solely responsible for this effect. In [29], the interested reader will find that macroscopic polymer scaling arguments can be successfully employed to estimate Mc as a function of σ c , reaffirming that this effect is the result of polymer-level confinement effects in the different voids. Although MC simulations support the conclusions from MD, kinetic effects in the latter are still a concern.

This is especially true in light of the so-called ‘rule of stages’, which states that crystallisation from a supercooled solution is expected to proceed via a sequence of intermediate metastable states [36–38]. That is, one might expect the colloids to initially form the less stable HCP crystal before slowly annealing into the FCC polymorph [28]. To investigate if this effect may be responsible for the widespread occurrence of HCP crystals in MD simulations, nearly spherical defect-free crystallites of both polymorphs were each placed in a polymer bath at the same conditions, which were found to produce the HCP polymorph out of an initially disordered state of colloids. MD simulations were then performed, and the crystal was analysed over time using Steinhardt–Nelson bond order parameters [39,40], which accurately reflect what fraction of the colloids are present in each polymorphic form (cf. Figure 3(d)). When the colloids are initialised in a HCP configuration, the crystal remains as this polymorph. However, if the crystal is initialised from a FCC configuration, it anneals into the HCP crystal over the

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Figure . Polymorph stabilisation with asymmetric tetramers. Reproduced from []. (a) Difference in polymer excess chemical potenex tial between close-packed polymorphs, μex /kB T = (μex FCC − μHCP )/kB T , for an adsorbed tetramer with beads of different sizes. Each bead was modelled as a hard sphere connected to its neighbours by square-well bonds with zero energy within % of the reported bond lengths, but infinite elsewhere. This greatly accelerated the simulations which were performed over a fine grid of different bond lengths between the interior beads, rmid , and between an interior and terminal bead, rend . (b) At rmid = , two different rend values, indicated by arrows in part (a), were considered in detail. MD simulations were performed using continuous potentials which mimicked their discontinuous counterparts from the MC simulations, and the resulting colloid–colloid radial distribution functions are shown here.

same amount of simulation time [29]. Therefore, the HCP crystals observed in these simulations are not metastable intermediates, but truly the most thermodynamically stable polymorph in these binary colloid–polymer systems.

3. Polymer morphology controls relative stability If the relative stability of the FCC and HCP crystals is based on the distribution of free volume inside the competing crystal forms, then it is plausible that by engineering certain aspects of the polymer, one may be able to achieve the opposite result and stabilise the FCC crystal instead. For instance, consider the asymmetric tetramer depicted in Figure 4(a). This linear polymer contains four beads; the two terminal beads are identical in size and larger than the two interior beads, which are also identical in size to each other. The relative location of the two large ends can be controlled by modulating the bond lengths between the interior beads, rmid , and between the interior and terminal beads, rend . This toy model is analogous to a block copolymer where chemically different blocks, interacting with the background solvent, experience different effective solvent qualities [29]. MC simulations were repeated for this tetramer when confined in each polymorph to measure the chemical potential difference between the two at many different rmid and rend values [29]. The interior beads had a diameter of unity, while the terminal beads had a diameter of 1.75. With this diameter, the large ends only fit inside the OVs when the colloid diameter is σ c = 6.45. Therefore, changing the relative distance between the tetramer’s ends amounts to controlling the optimal separation distance of the OVs.

Figure 4(a) depicts the differential chemical potential surface of the polymer. By convention, a positive μex implies the HCP polymorph is more stable than the FCC, and vice versa. Indeed, the surface displays rich behaviour with peaks and valleys which suggest domains where each polymorph could be stabilised over the other by up to 4 kB T per adsorbed tetramer. Specifically, when rmid = 2, increasing rend from 2.5 to 3.5 is predicted to change the relative stability of the polymorphs from favouring the FCC to the HCP. Figure 4(b) illustrates the final structure of the colloidal crystal obtained from MD simulations at a sufficient overall density to induce crystallisation [29]. As predicted by MC, the smaller value of rmid leads to the selective formation of the FCC crystal, whereas the HCP is clearly formed in the case of the larger rmid . Increasing the tetramer’s density was found to increase the bias and accelerate the annealing of transient defects, since additional tetramers adsorb in the crystal, increasing the overall magnitude of the bias [29]. This toy model demonstrates that by carefully controlling the size and shape of the adsorbed polymer, the relative stability of the two close-packed polymorphs can be tuned at will. While the beads on this polymer are asymmetric, the chain is still linear. By virtue of the finite length of the tetramer chain, the relative polymorph stability still depends on the spatial correlation between a pair of voids that the large ends inhabit, though in this case the two voids need not be the nearest neighbours. It is plausible that polymers with more complex architectures could impose correlations between more than two voids in the crystal. Consider the case of a star polymer: these polymers are composed of f linear chains, called ‘arms’, joined at a common central endpoint. In the canonical case

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Figure . Relative stability of the FCC and HCP polymorphs in the presence of f =  star polymers. Reproduced from []. (a) In the top panel, the difference in the total excess chemical potential of the star polymer between the two polymorphs is shown, following the convention of Figure , when σ c /σ m = . The lower panels illustrate the ensemble-averaged number of arms in each polymorph, N 〉, that terminate in void type B when the centre of the star is located in void type A (A–B). Lines serve as a guide to the eye. Marm =  corresponds to when only the star’s centre is present. (b) Representative snapshots of a star polymer’s configuration when its centre is located in each void type for both close-packed polymorphs as Marm increases. The image depicted is for when σ c /σ m = ., which is qualitatively identical to the case of larger asymmetry. The relative size of the colloids has been reduced for visual clarity, and the most stable resulting configuration is indicated by a dark outline; for small stars, the OVs of the FCC and HCP are equally stable environments, so both are lightly outlined. (c) The most stable close-packed polymorph in the presence of a star polymer with a given degree of polymerisation, Marm , at a given solvent quality, expressed in terms of the Flory–Huggins χ parameter. Here σ c /σ m = , as in part (a).

considered here, each of these polymers are identical, linear homopolymers with Marm beads. A linear homopolymer may be considered to be a star with f = 2 arms. Mahynski et al.[31] explored the capacity of star polymers with a small number of arms to impose a correlation between more than two voids in the crystals. Once again, monomer bonds were described by Equation (2) and all particles were purely repulsive as described by Equation (1). In this study, it was necessary to increase the size asymmetry of the monomers and colloids such that σ c /σ m = 11. Stars with f = 3 arms are the focus of the discussion here, as they represent the simplest perturbation to the case of linear homopolymers (f = 2). Incrementalgrowth MC simulations were repeated for star polymers, as before, by cyclically increasing the length of each arm by one bead. At the end of each cycle, each arm’s length was identical and increased by one over the previous cycle. Cycles were then repeated to continuously increase the size of the star, and the total excess chemical potential of the star was obtained by accumulating the sum of

the incremental chemical potentials of each new bead on each arm. The top panel of Figure 5(a) depicts the difference in the total excess chemical potential of an adsorbed star polymer between the FCC and HCP polymorphs. Once again there is no significant difference between the crystals for small polymers, but as Marm increases the difference becomes positive, indicating a bias towards the HCP polymorph. The same results were obtained with linear homopolymers, but in this case the maximum in μex /kB T is quickly followed by a minimum which falls below zero. This indicates a sudden reversal of the relative stabilities, not found when f = 2. Examining the location of the terminal monomers on each arm relative to an adsorbed star’s centre during the MC simulations gives insight into the cause of this. The lower panels of Figure 5(a) show the ensemble-averaged number of arms whose terminal monomer is located in void type B, while its central monomer is located in a void of type A (A–B). For instance, the red curve indicates that when

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Marm > 0, the central bead and all f = 3 arms are located in octahedral voids (OV–OV). In fact, the arms and core are usually in the same OV. However, for a star inside the FCC polymorph, when Marm > 12, the blue curve exceeds the red, indicating it is now more common to observe a star with its core inside a TV and its arms inside OVs (TV–OV), than an OV–OV configuration. No such effect is observed in the HCP crystal at this point, which corresponds to where the relative stability of the two polymorphs reverses. For stars of this size, the centre migrates from a larger OV to a smaller TV. This curious effect results from competition between the entropic cost of confinement for the star’s core versus that of its arms (henceforth collectively referred to as the ‘corona’) [31]. Constrained MC simulations were repeated such that the core of the star was forced to exist in each void type for each polymorph as Marm was increased, in order to estimate the relative free energy of each type of configuration. Changes in the relative free energy of each constrained simulation were shown to be in good agreement with unconstrained MC simulations which did not restrict the star’s core [31]. The trends are qualitatively summarised in Figure 5(b), where the lowest free energy state of the system is outlined by a box. When the star is very small, it tends to inhabit the OVs regardless of the polymorph it is adsorbed in. Thus, there is no significant preference conferred by the star, indicated by the two light boxes. As the star’s corona grows, the HCP polymorph is stabilised over the FCC due to the connectivity of the large OVs in that polymorph, which is not found in the FCC. This is identical to the effect observed when f = 2. As the size of the corona increases further, the emergent stability of the FCC crystal, and corresponding migration of the star’s core from an OV to a TV, may be understood geometrically. In the FCC crystal, each OV is surrounded completely by TVs. Thus, a star with its core adsorbed in a FCC crystal’s OV has the highest free energy of those depicted in Figure 5(b), because there are no OVs available to the corona. This is followed by a star whose core is adsorbed in a HCP crystal’s OV, which allows its arms to access its two neighboing OVs. If the core is confined to a TV in the HCP polymorph, there are a total of three neighbouring OVs in which the arms may be placed, making this state even more stable. The star’s configuration when its core is confined to a TV in the FCC and HCP crystals look identical in Figure 5(b); however, because of the stacking pattern of the FCC crystal, an additional fourth OV is available out of the plane of the image making this the lowest free energy state. This pseudo-spherical symmetry of the OVs around each TV is found only in the FCC crystal, and significantly increases the locally accessible free volume available to each of the arms. Although there is an

entropic cost to confine the core to the smaller TV, it is outweighed by the gain resulting from the corona’s ability to expand into large OVs. The overall size of the star polymer dictates the relative importance of the core and corona’s contribution to the star’s entropic confinement cost [31]. Therefore, by controlling the size of the polymer, it is possible to tune the relative stability of the FCC and HCP crystals when stars have adsorbed. One way this control may be exerted is by tuning the solvent quality. MC simulations were repeated in which the excluded volume between nonbonded monomer segments was progressively reduced by decreasing the value of  in Equation (1) [31]. When  ࣈ −σ m , there is essentially no excluded volume between such monomers, reminiscent of θ-solvent conditions. Whereas when  = 0, the monomers have a selfavoiding character akin to good solvent conditions. This may be expressed in terms of the Flory–Huggins χ = (1 − v* )/2, where v ∗ = (σm + )3 /σm3 is the excluded volume [31,41]. Figure 5(c) illustrates the effect of tuning χ for stars at a fixed chain length when σ c /σ m = 11. For large stars, reducing the solvent quality from good (χ = 0) to θ-conditions (χ = 0.5) at a fixed degree of polymerisation, Marm , leads to a transition from the stabilisation of the FCC to the HCP polymorph. It should be noted that in order to observe these effects, the star itself must be free to diffuse between the interstices of a crystal. Consequently, when σ c /σ m is small and f is large, the star cannot diffuse between the trigonal gaps. This is expected to lead to kinetic limitations, which must be carefully considered in any experimental attempt to realise this effect. This also manifests in MC simulations in which ergodicity issues arise as a consequence of artificially constraining the star to the void it initially began sampling when it had a smaller corona. This was carefully considered in [31] and is the reason that most results reported there were obtained at larger σ c /σ m ratios than those studied in previous investigations.

4. Energetic effects With the exception of the solvent-mediated control over star polymers just discussed, the relative stability of the FCC and HCP polymorphs in the presence of adsorbed polymers has, thus far, been purely the result of entropic considerations. However, energetic effects, such as dispersion interactions, are often significant under experimental conditions. Of particular concern is the impact that attractive interactions between monomer segments and colloids could have on this polymorph stabilisation mechanism. In [30], this was investigated by modelling the monomer–colloid interaction via a hard-sphere

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9

Figure . Consequences of attractive interactions between monomers and colloids on the relative stability of the close-packed polymorphs with adsorbed linear homopolymers. Reproduced from []. (a) Difference in the total excess chemical potential of a single adsorbed chain of length, M, between the FCC and HCP polymorphs for various B∗2 values. The difference follows the same convention in Figure . (b) A representative snapshot of a chain of length M = , when σ c /σ m = ., moving between pairs of TVs in a HCP crystal when B∗2 = −1.50. Some of the HCP crystal’s TVs and one OV are traced out to give perspective. The relative size of the colloids with respect to the monomer beads has been reduced for visual clarity.

potential with a Yukawa-like tail: Um,c (r) = ⎧ ∞ ⎪ ⎪   ⎨ +1−r  exp − ⎪ r− κ −1 ⎪ ⎩ 0

r ≤+1  + 1 < r ≤ rcut +  r ≥ rcut + t (6)

For the cases discussed here, κ −1 = 1/5, σ c /σ m = 6.45, and the attractive interactions were cut off at rcut = 2 [30]. The reduced second virial coefficient may be computed from this cross-interaction, which gives a measure of the effective excluded volume that exists between a monomer segment and a colloid, relative to the case when they are both hard spheres: B∗2 = −

3 (1 + )3

0

rcut +

   exp −Um,c (r)/kB T − 1 r2 dr



(7) Once again, bonds between monomers were modelled using Equation (2), and MC simulations were repeated to assess the relative stability of the polymorphs when a single polymer has adsorbed. The purely repulsive nature of the monomer–colloid interaction given by Equation (1) yields a value of B∗2 = 1.01 ≈ 1, which has been constant until this point in the discussion. Consequently, the colloids have served to delineate the bounds between the different void types for the polymer rather effectively. However, if the value of B∗2 is reduced to zero, the concept of ‘voids’ inside the crystal phases becomes more difficult to

justify. Since the monomers and colloids have effectively no excluded volume, it is questionable whether or not the polymer can detect any difference between different interstitial void spaces (OV vs. TV). The red curve for B∗2 = 1.01 in Figure 6(a) is identical to the information found in Figure 3(c). It displays the characteristic positive peak for large M values, indicating the HCP crystal is a lower free energy environment for the linear polymer, while for smaller chains, there is no significant difference between the polymorphs. A value of ε/kB T = 2.78 in Equation (6) yields B∗2 = 0 in Equation (7). Figure 6(a) reveals that under such conditions, there is no polymorphic bias imposed by the polymer. The lack of excluded volume effectively eliminates the polymer’s capacity to sense the voids, hence the absence of any bias. Increasing the magnitude of the cross-interaction between the monomers and colloids to ε/kB T = 4 in Equation (6) results in B∗2 = −1.50. Given the initial trends in Figure 6(a), it may be expected that this would result in μex < 0, favouring the formation of the FCC polymorph. To the contrary, Figure 6(a) reveals that the difference in an adsorbed polymer’s total excess chemical potential between the crystals is positive and oscillatory. Once again, the reason for this may be traced back to differences in the spatial distribution of the void types between the polymorphs. When the monomer–colloid interaction is strong, the smaller TVs provide regions of higher energy density than the larger OVs. As a result, the monomers preferentially adsorb in the TVs. However, non-bonded monomers are still repulsive and segments will quickly fill a single TV. The HCP polymorph fortuitously provides a polymer adsorbed in a TV with direct access to a neighbouring TV, whereas all

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N. A. MAHYNSKI

TVs in the FCC crystal are surrounded by OVs. Analysis of the void occupancy reveals that the oscillations in μex are a result of this [30]. In the FCC crystal, a polymer will initially fill a TV, then expand through a neighbouring OV to reach the next TV as quickly as possible. In the HCP polymorph, twice as many monomers can be placed in TVs before having to traverse an OV, after which the polymer reaches another TV pair. This scenario is depicted in Figure 6(b). Intriguingly, this reveals that even when energetically favourably cross-interactions between the monomers and colloids exist, polymers always tend to favour the HCP crystal over the FCC. In the entropically dominated case, when B∗2 > 0, adjacent OVs interact with the polymer to stabilise the HCP polymorph, whereas in the energetically dominated case, when B∗2 < 0, adjacent TVs also stabilise the HCP [30]. This is diametrically opposed to the case when a pure colloidal suspension crystallises, in which the FCC is always, if only marginally, more stable than the HCP crystal.

5. Conclusions and outlook Summarily, polymer internal degrees of freedom play an important role in determining the most stable crystal structure in nanoscale binary colloid–polymer mixtures. A polymer’s architecture and interparticle interactions may be intelligently tuned to strongly bias the formation of either the FCC or HCP polymorph. In the cases discussed here, the free energy difference between the two structures could be amplified by over three orders of magnitude. This is a direct consequence of the different distribution of interstitial free volume between the crystals. The finite size of a polymer implies that when it partitions into the crystal phase it can sense differences in the locally accessibly free volume if the polymer’s overall size is commensurate with the length scale over which these void asymmetries exist. When the polymer–colloid interactions are purely repulsive, this polymorphic bias is entirely a result of entropic considerations. Since this mechanism of crystal selection is entropic in nature, it is potentially amenable to simultaneous deployment with other energetically driven ‘bottom-up’ synthesis strategies [42]. Recent progress has been made illustrating that a combination of these approaches can be used to achieve very fine control over more complex crystals than close-packed ones. In fact, the principles outlined here may be fruitfully extended to bias lower density crystals against competing polymorphs, provided measures are taken to prevent their collapse into close-packed structures [33]. This reinforces the generality of this polymorphic selection mechanism, which is consistent with its simple, but widely overlooked, origins.

The caveat to achieving this thermodynamic bias is that the polymer must appreciably partition itself into the colloidal crystal. For example, if the maximum preference for the HCP crystal conferred by a linear homopolymer is approximately 2kB T per polymer adsorbed (cf. Figure 3), then a single adsorbed polymer is sufficient to counteract the inherent bias towards the FCC crystal (of about 0.001kB T per sphere) in a crystallite composed of roughly 2000 colloids. The magnitude of the overall bias is clearly proportional to the number of polymers that adsorb in the crystal phase, which usually increases with the overall polymer density. Presumably, the maximum bias would be achieved when all OVs are filled with polymers. When each adsorbed polymer is confined to its own OV, different polymers are not expected to interact significantly, which is consistent with observations made in the molecular dynamics studies discussed in Section 2 [29]. Therefore, it is reasonable to continue to treat the polymers as if they are non-interacting. This is ultimately why predictions made by MC simulations containing only a single polymer seem to accurately reflect the behaviour found in MD where many polymers are adsorbed in the crystal. However, if the polymer osmotic pressure is very large, it is expected that the crystal’s interstices will eventually saturate and polymers may begin to interact; the polymorphic preference induced by ‘dilutely’ adsorbed polymers may begin to suffer as a result. Such investigations are the subject of ongoing work, which suggest that indeed, the resulting behaviour becomes significantly different [32]. Nonetheless, it is clear that often there exists a reasonable range of polymer densities which can both induce crystallisation of the colloids and lead to sufficient polymer partitioning to bias the formation of a desired polymorph, without incurring such complications. The basic principles I have outlined here also lay the groundwork for future experimental realisations of this phenomenon. Colloidal diameters must be on the order of several hundred nanometers to produce close-packed crystals with optical characteristics. Thus, polymers must have large persistence lengths so that their Kuhn segments are commensurate with the size of the monomer beads in Figure 3. Double-stranded DNA is one candidate, though others may also be contrived [29]. Solvent-mediated control, demonstrated in Section 3 for star polymers, also offers a convenient platform to begin with, since χ may be controlled via experimentally accessibly parameters such as temperature. This could provide an in situ ‘switch’ to reverse the relative stability of competing polymorphs, which may be more difficult to achieve with systems such as the asymmetric tetramers, presented in the same section, due to kinetic considerations. This polymorphic selection mechanism represents a new addition to the ‘engineering toolkit’ with

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which well-ordered colloidal crystals can be designed. Future work will continue to focus on extending the utility of this entropic control platform in order to stabilise arbitrarily chosen crystals, and establishing plausible experimental routes to these ends [32,33].

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Acknowledgements I would like to acknowledge A. Z. Panagiotopoulos and S. K. Kumar for many helpful discussions in this subject area. In addition, I would like to thank the US National Science Foundation for the financial support of a Graduate Research Fellowship under grant CBET-1402166 at Princeton University, and the US National Research Council for support from a postdoctoral research associateship at the National Institute of Standards and Technology.

Disclosure statement No potential conflict of interest was reported by the author.

Funding I would like to thank the US National Science Foundation for the financial support of a Graduate Research Fellowship at Princeton University [grant number CBET-1402166]; the US National Research Council for support from a postdoctoral research associateship at the National Institute of Standards and Technology.

ORCID Nathan A. Mahynski

http://orcid.org/0000-0002-0008-8749

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