C. Patrick
Royall
*^{abcd},
Wilson C. K.
Poon
^{e} and
Eric R.
Weeks
^{f}
^{a}HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK. E-mail: paddy.royall@bristol.ac.uk
^{b}School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK
^{c}Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol BS8 1FD, UK
^{d}International Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan
^{e}SUPA and School of Physics & Astronomy, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
^{f}Department of Physics, Emory University, Atlanta, GA 30322, USA

Received
30th May 2012
, Accepted 20th September 2012

First published on 9th October 2012

We recently reviewed the experimental determination of the volume fraction, ϕ, of hard-sphere colloids, and concluded that the absolute value of ϕ was unlikely to be known to better than ±3–6%. Here, in a second part to that review, we survey effects due to softness in the interparticle potential, which necessitates the use of an effective volume fraction. We review current experimental systems, and conclude that the one that most closely approximates hard spheres remains polymethylmethacrylate spheres sterically stabilised by polyhydroxystearic acid ‘hairs’. For these particles their effective hard sphere diameter is around 1–10% larger than the core diameter, depending on the particle size. We argue that for larger colloids suitable for confocal microscopy, the effect of electrostatic charge cannot be neglected, so that mapping to hard spheres must be treated with caution.

Early hard sphere experiments include Bernal's use of ball bearings to model liquid structure.^{13} However, ball bearings have negligible thermal motion. Suspensions of mesoscopic colloids exhibit Brownian motion and are thermodynamically equivalent to atoms and small molecules.^{14} In 1986, Pusey and van Megen^{15} showed that a suspension of sterically stabilised polymethylmethacrylate (PMMA) particles showed hard-sphere-like equilibrium phase behaviour. Their work led to many experimental studies of the statistical physics of hard spheres using colloids as models. Since Pusey and van Megen's work, the equation of state of hard-sphere colloids has been determined,^{16} crystal nucleation has been observed,^{17,18} and the glass transition has been studied.^{19}

The body of experimental research just reviewed relied on light scattering as the structural and dynamical probe. The advent of single particle tracking in real space with confocal microscopy^{20,21} opened a new dimension in experiments on hard-sphere-like systems, yielding an unprecedented level of detailed information.^{22} Confocal microscopy of hard-sphere-like suspensions is thus ideal for studying generic processes where local events are important, such as crystal nucleation,^{23} melting^{24} and dynamical heterogeneity.^{25,26}

In principle, the thermodynamics of a system of hard spheres is controlled solely by the state variable ϕ. We have recently reviewed the experimental determination of volume fraction^{27} and concluded that, although relative values of ϕ may be known with high precision, absolute values can only be determined to within 3–6% accuracy. This matters, especially when dealing with dynamical properties (e.g., phase transition kinetics and the glass transition), since these can be very strong functions of ϕ.

But the accurate determination of ϕ is only part of the experimental challenge. The other part of the challenge was hinted at by the title of Pusey and van Megen's 1986 paper, “Phase behaviour of concentrated suspensions of nearly hard colloidal spheres”,^{15} where we have added the italics to emphasise the point in question, namely, that true hard spheres do not exist in reality. In Pusey and van Megen's case, the lack of hardness is almost certainly due to the small but finite compressibility of the PHSA stabilising ‘hairs’. The same is generically true of other sterically stabilised particle systems.

In very-nearly-hard, sterically stabilised suspensions, a new sample has to be made for every state point ϕ. Apart from being cumbersome, this also restricts the accuracy with which the sole thermodynamic control parameter can be ‘tuned’. Thus, there has been a drive to use particles such as ‘microgels’, whose diameter is temperature dependent. Since temperature, T, can be tuned far more accurately than ϕ, this allows the scanning of a single sample very finely through ϕ space by varying T. But the price paid for the ‘tunability’ of particle diameter is that some softness is built in by design.

To understand a final reason why real model hard-sphere colloids may be somewhat soft, consider why in practice, not only ϕ but the particle diameter σ matters. Experimentally, colloids as synthesised are usually not matched in density to the dispersing medium. Significant sedimentation (or, less usually, creaming) during the timescale of an experiment therefore presents a problem. In light scattering studies, this is circumvented to a large extent by the use of small particles, say, σ ≲ 400 nm, so that for PMMA particles dispersed in cis-decalin, a single particle sediments ≲1 mm per day. The middle part of a bulk sample would therefore be little affected by sedimentation over a day. But if one wants to eliminate the effect all together (e.g. for long-time measurements), or when larger particles (say, σ ≳ 1000 nm) are needed for imaging purposes, then solvent mixtures for density matching are needed, which often introduces significant charging. The practical need for larger particles therefore inevitably brings softness.

In this second part of our review, we survey critically these three sources of softness in experimental systems of colloids that have been used to model hard spheres, Fig. 1. In the spirit of the first part of our review,^{27} we seek to provide a means by which the hardness may be assessed, and consider the consequences this may have on the behaviour of the system. Likewise, we argue for more clarity on the part of experimentalists, concerning the softness of the systems they use. We also suggest a number of criteria by which experimentalists (and theorists using experimental data) may judge whether a certain system of colloids may be considered ‘hard enough’ for answering particular physics questions.

Fig. 1 Schematic representation of various models for hard-sphere colloids. (a) Sterically stabilized particle, with surface ‘hairs’ (not to scale), where the average thickness of the stabiliser layer and the core–shell diameter σ_{cs} = σ_{c} + 2 are needed for a full characterisation. (b) Microgel particle, which is a heavily cross-linked polymer. (c) Charged colloid, where the electrical double layer (shaded) gives rise to an effective diameter σ_{eff}. |

Below, we first discuss mapping experimental systems onto hard spheres via measuring the inter-particle potential in Section 2. We then treat softness of non-electrostatic origins in Section 3 and of electrostatic origins in Section 4; in general, both effects are present simultaneously. Finally we give a worked example in Section 5, illustrating many of the points raised throughout this review, before concluding in Section 6.

First, however, we briefly review how a knowledge of u(r) can be used to determine an effective hard-sphere diameter, σ_{eff}. Perhaps the simplest approach is to set an effective hard sphere diameter σ_{kT} such that the inter-particle repulsive energy at this centre-to-centre separation between two particles is equal to the thermal energy, i.e.

βu(r = σ_{kT}) = 1, | (1) |

(2) |

Other, yet more sophisticated mappings, exist, such as that due to Andersen et al.;^{29} this approximation is known to work well for mapping the static properties of liquids and more recently structural relaxation time near the glass transition.^{30} All of these approaches rely on knowing u(r). We now review methods for gaining this knowledge.

In the case of PMMA particles sterically stabilised by poly-12-hydroxyl steric acid (PHSA) ‘hairs’, the interaction may be inferred via the direct measurement of the interaction potential between mica surfaces coated by PHSA using the surface-force apparatus. The results were well described by an inverse power law, suggesting a reasonably (but not absolutely) hard interaction;^{31} see Section 3 for more details.

Other methods for measuring colloidal interactions directly include total internal reflection microscopy, which measures the force between a colloid and a glass wall,^{32} and atomic force microscopy with a colloid on the tip of the cantilever.^{33} The interaction between two non-index-matched colloids confined to a line can be measured by optical tweezers.^{34}

One attraction of such direct methods is that no a priori assumption need be made about u(r). However, their use requires care. Thus, e.g., in the case of optical tweezers, relatively small and subtle experimental errors can lead to the wrong sign of the interaction between charged colloids.^{35}

Inverting the real-space pair correlation function, g(r), can also give the interaction potential,^{39,40} and is subject to the same ambiguities, especially when many-body effects are present.^{41} But if g(r) can be measured in the limit of vanishing ϕ, then the inversion to give u(r) reads:^{36}

(3) |

This approach has become possible with the advent of real-space techniques, which allows the determination of g(r) by direct counting of particles.^{21,40,42,43} In the limit represented by eqn (3), monodisperse hard spheres give a perfect step function form for g(r), but polydispersity and particle tracking errors would blur the sharpness of the edge at r = σ, as would a small degree of softness.^{43} In Fig. 2, we show the g(r) measured in this limit for a putative hard-sphere suspension.^{44} It is immediately clear that these particles cannot, in fact, be hard spheres. The peak in the dilute-limit g(r) is due to a short-range inter-particle attraction.† Computer simulations can be used to fit the measured dilute-limit g(r) using a square-well attraction for u(r); importantly, however, a distribution of particle sizes as well as random tracking errors (both modelled by Gaussians) are essential to obtain a good fit. Interestingly, it has been argued^{45} that eqn (3) remains a remarkably accurate approximation at finite but modest ϕ.

Fig. 2 Attractions in “hard” spheres for confocal microscopy. Experimental data are from ref. 44. The dashed black line is from Percus Yevick theory for hard spheres at ϕ = 0.071.^{36} The solid line is computer simulation data with particle tracking errors and polydispersity added, for a square well attraction of depth k_{B}T and range 0.09σ. |

(4) |

For hard spheres, B^{HS}_{2} = 2πσ^{3}/3. Thus, an effective hard-sphere diameter for a slightly soft system can be obtained by taking

(5) |

Note that even if the value of σ_{virial} obtained in this way corresponds very closely to, say, a core radius determined from, e.g., electron microscopy, it does not follow that the system approximates closely to hard spheres: B_{2} is an integral quantity, so that a balance of repulsion and attraction can masquerade as a good fit to hard spheres on this level without u(r) being actually a hard potential.

(6) |

The relative range of u_{s} depends on the particle size, for example n = 170 for particles with a hydrodynamic diameter of σ_{H} = 200 nm, and increases with particle diameter. Likewise the strength of the interaction also depends on the particle size, with u_{s}(σ_{H}) = 146 k_{B}T for σ_{H} = 200 nm. The results of Bryant et al.^{31} are replotted in Fig. 3. These quantify what is intuitively obvious, namely, that for a fixed length of stabilising ‘hairs’, larger particles are relatively harder.

Fig. 3 Estimation of effective colloid–colloid interactions in sterically stabilised PMMA particles: (a) σ_{H} = 200 nm, (b) σ_{H} = 2000 nm. In both parts, light pink lines denote u_{s}(r), the interaction due to the steric stabilisation. The dashed blue line in each case represents unscreened weak electrostatic interactions in low dielectric constant solvents (cis-decalin/TCE) calculated for effective charges Z = 2 and Z = 16 for (a) and (b) respectively and a Debye length of κ^{−1} = 5000 nm. The solid blue line represents typical screened electrostatic interactions: (a) in water (charge Z = 1700 and Debye length κ^{−1} = 4 nm), and (b) a density matching mixture of cis decalin and CXB (charge Z = 500 and Debye length κ^{−1} = 100 nm). |

Note that despite the steepness of the potential represented by eqn (6) with n ∼ 10^{2}, the effective volume fraction of PMMA colloids and other similarly sterically stabilised particles can nevertheless exceed the limit of random close packing, since the stabiliser layer can be compressed. Thus, for example, centrifuging PMMA colloids almost invariably generates sediments that are, at least initially before any relaxation, at volume fractions beyond random close packing. This illustrates well the point that whether certain particles are ‘hard enough’ depends on the experimental context.

Few analytic expressions for microgel–microgel interactions exist. Expressions for polymer-covered flat surfaces have been suggested as a first approximation.^{50} The steric interactions between neutral microgels have been likened to crosslinked polymers, so that a form for the interaction can be obtained provided the density profile of the particle is known.^{51} Additional interactions in ionic microgels do have analytic forms,^{52} which can be added to the steric repulsion, for which an inverse power form is often assumed.

Irrespective of the absence of well-attested analytical forms for the inter-particle repulsion, swellable colloidal microgels are definitely not hard, and the degree of swelling depends sensitively on the experimental conditions, including possibly the concentration of microgel particles. If the latter is a significant effect, then any ϕ_{eff} would become state-dependent.

We now review a popular system, dispersions of poly(N-isopropylacrylamide) (PNiPAM) microgel particles,^{53} for which water is a poor solvent at T ≳ 33 °C, so that at and above this temperature, the diameter of the particles in an aqueous environment dramatically shrinks. Salt is usually added to screen electrostatic interactions, see Section 4. Changing the amount of the cross-linker N,N′-methylenbisacrylamide tunes how much shrinkage occurs.^{54} Sufficiently monodisperse samples crystallise at high volume fractions, giving crystals whose structure has been variously reported as face-centred cubic^{55} or as consisting of the more or less random stacking of hexagonally packed layers.^{56} Richtering and co-workers have examined the physical properties of PNiPAM microgel suspensions with a variety of probes and discussed their findings in terms of possible mapping onto hard spheres. We mention three aspects.

First, neutron scattering shows that an individual particle has a constant-density core, surrounded by a corona in which the density gradually decreases to zero over a distance that is approximately twice that of the core.^{57} From this finding alone, we expect the particles to be significantly soft.

Next, the structure factors of PNiPAM suspensions at progressively higher particle concentrations have been measured, and compared to those of hard spheres.^{58} Using two different methods of data analysis, it was concluded that the S(q) of these suspensions could be described adequately within a hard-sphere framework by assigning ϕ_{eff} (or, equivalently, σ_{eff}) to the particles at concentrations ϕ_{eff} ≲ 0.35. Above this concentration, increasingly large deviations from hard-sphere-like behaviour were observed. Interestingly, in a later paper,^{59} the same group points out that the mapping to hard-spheres at ϕ_{eff} was obtained by making one of two assumptions: either that a hard-sphere structure factor (from Percus–Yevick theory) in fact fitted the data, or that the form factor of a single particle^{57} determined in the low-concentration limit did not change when the microgel concentration was increased, which is not self-evidently true. Moreover, in the earlier paper, effective volume fractions determined from effective diameters obtained by fitting neutron scattering form factors do not match the volume fractions used for corresponding hard sphere structure factors in ref. 58.

Finally, Richtering and coworkers investigated the fluid-crystal coexistence gap.^{54,59} In one study of PNiPAM particles dispersed in water,^{54} a ϕ_{eff} was determined by requiring agreement with the hard-sphere expression for suspension viscosity at low concentrations, η/η_{0} = 1 + 2.5ϕ_{eff} + 5.9ϕ_{eff}^{2} (where η_{0} is the solvent viscosity). This procedure is problematic because of the aforementioned state-dependence of σ_{eff} (and therefore of ϕ_{eff}). Nevertheless, using this mapping, the fluid-solid coexistence gap was found to be ϕ^{f}_{eff} = 0.59 ≤ ϕ ≤ 0.61 = ϕ^{m}_{eff}, i.e. it occurs at significantly higher concentrations than that in hard spheres (0.494 = ϕ^{f}_{HS} ≤ ϕ_{HS} ≤ ϕ^{m}_{HS} = 0.545), and is substantially narrower (ϕ^{m}_{eff} − ϕ^{f}_{eff} = 0.02, ϕ^{m}_{HS} − ϕ^{f}_{HS} = 0.051).‡ Furthermore, the ϕ_{eff} obtained from the viscosity measurements does not match that obtained from fitting of the hard-sphere structure factors where by construction in the hard sphere fluid ϕ_{HS} ≤ 0.494. This shows nicely that softness effects influence static and dynamical properties in different ways. The discrepancy may be related to the fact that σ_{eff} for microgels is sensitive to the osmotic pressure, and thus can vary under shear, and also at different state points. Consequently, a mapping to hard spheres at a given state point under zero shear may not hold for other state points, or for the same state point under shear.

Comparison with simulations^{62} of the freezing of particles interacting via a power-law repulsion u(r) ∝ r^{−n} gives n ≈ 13 for PNiPAM particles with 240 nm ≲ σ_{H} ≲ 300 nm. Comparable results have obtained from analysis of rheological data^{63} in oil-based microgel systems^{60} and the PNiPAM system.^{54,64} This is considerably softer than the inter-particle potential found^{31} for sterically stabilised particles of comparable size (σ_{c} = 200 nm), which can be characterised by a power law with exponent n = 170.

A different mapping was used in a second study of fluid-crystal coexistence,^{59} in which the PNiPAM microgel particles are now dispersed in dimethylformamide (DMF, a good solvent chosen for refractive index matching). The freezing concentration of the microgel particles expressed in mass fraction, μ = m_{microgel}/(m_{microgel} + m_{solvent}), was converted to an effective hard-sphere volume fraction by a multiplicative factor, ϕ_{eff} = Sμ, where the ‘swelling ratio’ S was chosen to yield a freezing volume fraction of ϕ^{f}_{eff} = 0.494. Interestingly, this procedure gave ϕ^{m}_{eff} ≈ 0.55 for the point at which 100% crystallisation should occur, consistent with hard-sphere behaviour. This agrees with the findings by another group of a recent imaging study using larger PNiPAM particles dispersed in an aqueous medium,^{56} but contrasting strikingly with the previous finding by the same group of a significantly narrower coexistence gap.^{54} The same study found that the collective diffusion of these microgel particles dispersed in DMF and their hydrodynamic interactions could not be well described by any mapping to hard spheres.

These studies illustrate some of the difficulties associated with mapping microgels to hard spheres. An additional issue is that of polydispersity. Both softness in the inter-particle potential^{62,65,66} and polydispersity^{66,67} affect the width of the coexistence gap, so that the effect of these two quite distinct physical factors may be difficult to disentangle. Fortunately, microgels can be synthesised with polydispersities as low as ∼1% before swelling, so that perhaps the polydispersity effect can be neglected in the first approximation (cf. the very small effect 1% polydispersity on the miscibility gap of hard spheres^{67}).

A variant of the ‘canonical’ microgel has been synthesised and characterised by Ballauff and co-workers^{68,69} consisting of a hard polystyrene core onto which is grafted a network of cross-linked PNiPAM, so that the swollen shell has approximately the same dimensions as the core radius (≈50 nm). To determine ϕ_{eff}, a core volume fraction ϕ_{c} was first measured by conversion from the particle mass fraction using the density of polystyrene. The hydrodynamic diameter of the particles, σ_{H} was then determined using dynamic light scattering; this was later confirmed to be very close to the diameter of the outer corona visible in cryo-transmission electron microscopy (cryo-TEM) images,^{69} which also gave the core diameter σ_{c}, and the polydispersity of the core–shell diameter distribution (≈9%). Finally, using ϕ_{eff} = ϕ_{c}(σ_{H}/σ_{c})^{3}, the fluid-crystal coexistence gap at 21 °C was found at 0.483 ± 0.007 < ϕ_{eff} < 0.546 ± 0.007, which is, within experimental uncertainties, very close to the hard-sphere interval of 0.494 to 0.545.§

It is interesting to analyse these fluid-crystal coexistence measurements further. As Ballauff and coworkers^{69} have pointed out, if ϕ^{f}_{eff} is rescaled to exactly 0.494, then melting occurs at 0.556. This gives a coexistence gap wider than in perfect hard spheres. If this does not reflect experimental errors, then the situation is somewhat unusual – the most common ‘culprits’, polydispersity and softness in the repulsive potential, both narrow the coexistence gap [Fig. 4(a) and (b)].^{62,65,66} However, short-range attraction in the potential has the opposite effect of widening the coexistence gap if the polydispersity¶ is low enough.^{72–74} On the other hand, 9% polydisperse hard spheres should be at or beyond the experimental limit of crystallisation,^{75} probably due to the onset of multiple solid phase coexistence in the phase diagram,^{67,76} which requires long-range particle motion for fractionation. That crystallisation was still observed in these PNiPAM samples to give a coexistence gap wider than that of hard-spheres underlines the lack of complete understanding of microgel physics.

Fig. 4 (a) The theoretical phase diagram of hard spheres at different polydispersities, σ. F = fluid, S = (crystalline) solid; thus FSS denotes fluid–solid–solid coexistence. Replotted from Wilding and Sollich.^{67} (b) Phase diagram of hard-core Yukawa particles, from ref. 65 in the absolute volume fraction – Debye length (1/κσ) plane. Here the contact potential ε_{Y} = 8k_{B}T. In the case of zero Debye length, the hard sphere limit is recovered. Replotted from Hynninen and Dijkstra.^{65} |

More recently, Ballauff and co-workers studied in detail the rheology of a 17% polydisperse suspension of their core–shell particles at and near the glass transition (found to occur at ϕ_{eff} = 0.640), and compared their data with mode-coupling theory (MCT) calculations for hard spheres.^{77} At this polydispersity crystallisation was inhibited, but this introduced an extra level of complexity into calculating ϕ_{eff}. The authors relied on the fact that the high-frequency viscosity had been found to be relatively insensitive to polydispersity,^{78} and mapped the high polydisperse system onto a less polydisperse system in which ϕ_{eff} had already been calibrated according to the procedure explained above. A large measurement of agreement with MCT predictions has been found. We note in this connection that the comparison with MCT mostly relies on a relative measure of the distance to the glass transition ϕ_{g}: ε = (ϕ − ϕ_{g})/ϕ_{g}, so that the work is perhaps less vulnerable to systematic or statistical uncertainties in arriving at ϕ_{eff}, although the influence of softness on ε should certainly be investigated – to our knowledge, so far it has not been.

Very recently, the rheology of sterically stabilised, microgel and hard-core–microgel-shell particles has been compared.^{79} Qualitative differences have been found, with the yield strain exhibiting non-monotonic behaviour with respect to ϕ in the case of the sterically stabilised (solid) particles and monotonic behaviour otherwise. Furthermore, microgels have^{54,80} been studied at effective volume fractions above random close packing, which is impossible for true hard spheres.

The treatment of electrostatic interactions on the mean-field, or linearised Poisson–Boltzmann (PB), level is largely adequate for our purposes. This is especially true for non-aqueous systems,^{48,90} in which the energetic penalty of ionisation is high, so that ion densities are low, and multivalent ions can be safely neglected.

The linearised PB theory is incorporated into the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory^{91} to describe the interaction between charged colloids. The original DLVO potential consists of van der Waals (vdW) and electrostatic components. We are primarily interested in situations in which sterically stabilised particles become charged, so we will assume that the steric repulsion is adequate to render the vdW component negligible. Instead, we will consider an inter-particle potential consisting of a steric repulsion, u_{s}(r), and an electrostatic interaction, which in linearised PB theory has an Yukawa form, u_{Y}(r):

u(r) = u_{s}(r) + u_{Y}(r), | (7) |

(8) |

Here, the contact potential is given by

(9) |

λ_{B} = βe^{2}/(4πε_{0}ε_{r}), | (10) |

Very recently, it has been shown that in the case of ionic microgels, electrostatics can add a further level of complexity to the effective interactions. Changing concentration leads to a change in ionic strength which in turn couples to the particle size. This can lead to a significant change in effective hard sphere diameter as a function of particle concentration.^{94}

It seems that the degree of charging in many systems^{40,49,92,96,97} can be described by the rule of thumb Zλ_{B}/σ ≈ 6. Thus, the system used by Pusey and van Megen,^{15} sterically stabilised PMMA (σ ≈ 700 nm) in a mixture of cis-decalin and carbon disulphide (ε_{r} = 2.64, λ_{B} ≈ 20 nm), can be expected to be charged to some extent, as later work on a similar suspension seems to confirm.^{97}

Nevertheless, when the particles are small enough, such charging can often be ignored. Fig. 3(a) (dashed blue line) shows u_{Y}(r) for a charge of Z = 2 on the surface of σ_{H} = 200 nm particles (corresponding to Zλ_{B}/σ ≈ 6) in a dispersion medium with a Debye length of κ^{−1} = 5 μm. The measured steric repulsion for sterically stabilised PMMA particles of this size^{31} is also shown. It is clear that for all relevant length scales in this situation, u_{Y}(r) ≪ k_{B}T. These particles can plausibly be considered hard spheres.

Four commonly used halogenated solvents are cycloheptyl bromide (CHB), cyclohexyl bromide (CXB), tetrachloroethylene (TCE), and carbon tetrachloride.|| In these low dielectric constant solvents (e.g., ε_{r} = 7.9 for CXB), colloids suitable for confocal microscopy (σ_{H} ∼ 2 μm) acquire a charge of Z ∼ 100–500. The Debye length in a density-matching CXB–cis decalin mixture can run to microns as the ionic strength (due predominantly to solvent self-dissociation) can be as low as 10^{−10} M,^{95} which is much lower than the ionic concentration in pure water (10^{−7} M).

The charge on the colloid, although much lower than what can be expected on similar sized particles in water, is now almost unscreened, which can lead to very long-ranged and strong interactions (ε_{Y} ≳ 100 k_{B}T). These interactions can and do vary from sample to sample, as the ionic strength in CXB and CHB varies from batch to batch, and as a function of time.^{40,48,49} Colloidal crystallization has been found in some cases^{23} at ϕ ≈ 0.4 (as compared to hard spheres at ϕ^{f}_{HS} = 0.494), but at least one experiment saw crystallization at volume fractions as low as ϕ ∼ 0.01;^{95} often these crystals were body-centered-cubic (bcc) in contrast to hard-sphere crystals which are random-hexagonal-close-packed (rhcp) or fcc. Furthermore, for some batches of CXB, the colloid charge can change with particle concentration, leading to strongly ϕ-dependent interactions, and even to re-entrant melting.^{48} The use of PMMA particles in halogenated solvents for confocal microscopy is therefore problematic.

The situation can be improved somewhat by the use of salts to screen the charges. The problem with this approach is that salts soluble in these solvent mixtures such as tetrabutyl ammonium bromide (TBAB)^{95} are soluble only to around 260 nM.^{48,49} This results in a Debye length of κ^{−1} ≈ 100 nm. Although this is substantially less the diameter of imageable colloids (σ_{H} ≳ 1 μm), it is not negligible and a noticeable degree of softness will likely result, Fig. 3(b). On the other hand, since the majority of ions now come from the salt, the ionic strength and therefore the colloid–colloid effective interactions will likely be reasonably independent of ϕ.

Another possibility is to use lower dielectric constant solvents such as TCE and CCl_{4} (ε_{r} = 2.5 and 2.24 respectively). Lower ε_{r} increases λ_{B}, eqn (10). The rule of thumb for estimating the degree of charging, viz., Zλ_{B}/σ_{c} ≈ 6, therefore predicts a lower Z. However, both of these solvents are strongly absorbed by PMMA, and can lead to a volume swelling of ≳40%.^{44} Unless the swelling is very closely monitored and characterised, it becomes a source of potentially large systematic errors,^{27} because ϕ ∝ σ^{3}. Solvent absorption also changes the density and refractive index of the particles. Thus, one of the initial attractions of using such halogenated solvents is lost – without swelling, adding one of these solvents can density match nearly exactly but also (fortuitously) nearly match the refractive index. With significant absorption, more TCE (say) than is needed for index matching has to be added to achieve density matching. Unless a third solvent is used to re-achieve index matching (which itself may lead to further swelling), a turbid sample results.

Such turbidity not only degrades image quality, but can also give rise to significant vdW attraction. For example, Fig. 2, which shows the measured g(r) of a ϕ = 0.071 suspension of sterically stabilised PMMA particles in a density-matching mixture of cis-decalin and TCE.^{44} The pronounced peak at touching immediately alerts us to the presence of inter-particle attraction, as simulations confirm.

Consider charged hard spheres, so that u_{s}(r) in eqn (7) is the perfect hard-sphere repulsion. We model scenarios that may reasonably represent sterically stabilised PMMA used in confocal imaging, and take σ_{c} = 2 μm, and a Debye screening length of κ^{−1} = 100 nm (so that κσ_{c} = 20). The latter is a round figure chosen to correspond roughly to a density-matching mixture of cis-decalin and CXB with the maximum possible amount of dissolved TBAB (260 nM). Consider two colloid charges Z = 500 and Z = 100. The former is consistent with the rule of thumb, Zλ_{B}/σ ≈ 6, while the latter is 5 times lower than predicted by this empirical relation. These charges have been reported in different studies^{40,48,98,99} for nominally identical PMMA particles used for confocal imaging dispersed in a density-matching mixture of cis-decalin and CXB. Eqn (9) gives ε_{Y} ≈ 10k_{B}T and ≈ 0.5k_{B}T for these two charges respectively.

Taking these parameters, the simulation results^{65} replotted in Fig. 4(b) can be used to determine freezing and melting for our hypothetical systems. These are delimited by red lines in Fig. 5 for (a) Z = 500 and (b) Z = 100. Note first that these values of not-very-large surface charge, variously reported in the literature for nominally very similar PMMA colloids, in fact pertain to rather large differences in the freezing/melting transitions, both in terms of absolute values of the transitions points and in terms of width of the coexistence region. Thus, some kind of ‘mapping’ is clearly necessary if we are to use either system to model hard spheres meaningfully.

Fig. 5 Mapping the phase behaviour of two hypothetical monodisperse charged hard sphere colloids (with parameters based on real systems – see text for details) to pure hard spheres. The hypothetical particles and solvent have the following properties: σ_{c} = 2 μm, κ^{−1} = 100 nm. The particles have two different charges (a) Z = 500, (b) Z = 100. In each case, differently shaded and delimited regions denote the fluid–solid coexistence gap of pure hard spheres (grey, ‘HS’), and from: simulations^{65} (red, ‘sim’), mapping using eqn (7) (lilac, ‘σ_{kT}’), and mapping using eqn (2) (blue, ‘σ_{BH}’). In (b), the result of mapping using eqn (7) does not change the coexistence region from that predicted by ‘sim’, and is not shown separately. |

We proceed to calculate σ_{eff} for the particles represented by these two parameter sets using either eqn (1) or eqn (2), and eqn (7)–(10). If mapping is to be helpful, we should expect that once we have transformed ϕ to ϕ_{eff}, freezing and melting of the two system should occur at approximately the hard-sphere values, viz., 0.494 and 0.545.

Fig. 5(a) shows that for the case Z = 500, eqn (1) predicts freezing at ϕ^{f}_{kT} = 0.484, just 0.010 from the ‘correct’ value.** However, the width of the coexistence gap is reduced, which is the result of the softness of the screened electrostatic (Yukawa) repulsion. Since the mapping involves scaling ϕ by constant, ϕ_{eff} = (σ_{eff}/σ)^{3}ϕ, it preserves the relative coexistence gap, so that the melting concentration, ϕ^{f}_{kT} is significantly underestimated. Conversely, the Barker–Henderson treatment gives a relatively accurate estimate of ϕ^{m}_{kT}, but rather significantly overestimates ϕ^{f}_{kT}. In the case of Z = 100, since ε_{Y} = k_{B}T, σ_{kT} = σ, so that eqn (1) maps perfectly onto hard spheres. In this case, the Barker–Henderson approach does the worse job, even when compared to the raw (unmapped) coexistence gap given by simulations of the bare Yukawa interaction. We note that since the coexistence gap varies with the interaction, mapping to the true hard sphere volume fraction at freezing inevitably gives an erroneous melting volume fraction unless the colloids are absolutely hard.

Fig. 5 therefore shows that the different methods of mapping to hard spheres do not give the same result. In practice, of course, the inter-particle potential is not exactly known, and polydispersity is inevitable. What is clear from the worked example summarised in Fig. 5 is that even in the ‘ideal’ case of monodisperse spheres with an exactly known inter-particle interaction, mapping to hard spheres is system- and approach-specific. In practice, of course, polydispersity introduces significant uncertainties, and the Debye length is often not determinable to high accuracy. Moreover, we stress that the values of Z = 500 and Z = 100 are taken from experiments on nominally identical systems. Thus, conclusions derived from any ‘mapping to hard spheres’, e.g. comparison of nucleation rates at nominally equivalent state points in the coexistence gap, must be treated with significant caution.

We end by making two further observations. First, we widen the scope of our enquiry from charged particles and microgels to other kinds of non-hard inter-particle interaction, and ask what requirements should be satisfied before one may fruitfully embark on the exercise of ‘mapping’ to hard spheres. We suggest the minimum conditions to be satisfied are:

(1) the absence of any attractive interaction, so that the equilibrium physics is dominated by entropic effects;

(2) crystals in a sufficiently monodisperse dispersion at high concentrations consist of the stacking of hexagonal layers.

The first criterion highlights the importance of refractive index matching to minimise the ubiquitous vdW interaction. The second criterion explains why charged hard particles with κσ ≳ 6, Fig. 4, and microgels^{55,56} are suitable candidates for mapping to hard spheres, but star polymers are not.^{81}

Secondly and finally, we point out that new developments in particle synthesis may yet produce μm-sized colloids that can be index and density matched using solvents or solvent mixtures that do not bring about charging and minimal swelling. Such development will be most welcome for the community of scientists wishing to use colloids to test fundamental theories of many-body physics via the hard-sphere model system.

Of course, soft particles, from microgels to star polymers and beyond, are fascinating systems in their own right.^{100} Furthermore, it is reasonable to enquire how much deviation from perfect hard spheres is acceptable, and the answer of course depends on what one wishes to study. Section 5 shows that the location of colloidal phase boundaries can vary in nontrivial and qualitative ways from hard spheres. However, it is plausible that slight softening may only have slight changes in, for example, the structural relaxation time.^{30}

Our point, then, is that while particle interaction details may not matter in some cases (such as the pair structure of dense liquids), there are plenty of cases where the behaviour of the system depends strongly on both the volume fraction ϕ and the interparticle interactions, therefore accurate knowledge of the both is essential. Crucially, the interactions may well be known to even less precision that the absolute volume fraction ϕ, which we have already argued is knowable only to 3–6%.

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## Footnotes |

† The step function form of g(r) for dilute hard spheres gives a featureless S(q) = 1. Residual attraction or softness therefore shows up much more obviously in real space. |

‡ The same narrowing of the fluid-crystal coexistence gap has been found in an oil-based systems of polystyrene microgels.^{60,61} |

§ We note that a similar system with silica cores and PMMA shells has been developed which has been density and index matched, and fluorescently labelled, making it suitable for confocal microscopy.^{70} |

¶ Experimentally, for hard spheres above a critical polydispersity, short-range attraction appears not to widen the coexistence gap.^{71} |

|| Note that carbon tetrachloride is a suspected carcinogen. |

** To put this difference in context, note that it is smaller than other sources of errors inherent in measuring ϕ.^{27} |

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