Maxence
Arutkin
^{a},
Elie
Raphaël
^{a},
James A.
Forrest
^{abc} and
Thomas
Salez
*^{abd}
^{a}Laboratoire de Physico-Chimie Théorique, UMR CNRS Gulliver 7083, ESPCI ParisTech, PSL Research University, 75005 Paris, France. E-mail: thomas.salez@espci.fr
^{b}Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
^{c}Department of Physics & Astronomy and Guelph-Waterloo Physics Institute, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
^{d}Global Station for Soft Matter, Global Institution for Collaborative Research and Education, Hokkaido University, Sapporo, Hokkaido 060-0808, Japan

Received
24th March 2016
, Accepted 6th June 2016

First published on 6th June 2016

Motivated by recent experimental results on glassy polymer nanoparticles, we develop a minimal theoretical framework for the glass transition in spherical confinement. This is accomplished using our cooperative-string model for supercooled dynamics, that was successful at recovering the bulk phenomenology and describing the thin-film anomalies. In particular, we obtain predictions for the mobile-layer thickness as a function of temperature, and for the effective glass-transition temperature as a function of the radius of the spherical nanoparticle – including the existence of a critical particle radius below which vitrification never occurs. Finally, we compare the theoretical results to experimental data on polystyrene from the recent literature, and we discuss the latter.

To our knowledge, the first report of anomalous dynamics in polymer nanoparticles is that of Sasaki et al.^{34} In that work, differential scanning calorimetry of aqueous dispersions of polystyrene nanospheres in the 21–274 nm radius range showed no evidence for a reduced glass-transition temperature, but instead revealed a radius-dependent value of the step in heat capacity at the glass transition. Analysis of the data suggested that this result is consistent with a near-surface region of size ∼3.8 nm not contributing to the transition. Then, Rharby used neutron scattering to measure mechanical deformations of polystyrene nanospheres individually dispersed in crosslinked polybuthylmethacrylate matrices, and deduced glass-transition temperatures that were reduced from the bulk value for nanospheres less than ∼30 nm in radius.^{35} Later measurements by Zhang et al.,^{38} and Feng et al.,^{40} showed reductions in the glass-transition temperature for spheres of larger radii, and with a strong dependence on the sphere coating. The latter fact is reminiscent of the strong effect even small amounts of residual surfactants can have on thin-film reductions in the glass-transition temperature, as reported by Chen and Torkelson.^{43} The above large disparity in observations may partially result from the much more difficult sample preparation in making dispersed nanospheres, as compared to thin films. Indeed, the presence of surfactants and/or residual monomers, and the uncertainty in final molecular weight could result in large variations between different experiments. Therefore, it appears necessary to establish a theoretical framework for the description of the glass transition in spherical confinement, and by this to enable comparisons between the observations in thin films and nanoparticles. As a remark, let us mention the existence of another theoretical attempt on nanoparticles, using a thermodynamical analogy between vitrification and crystallization.^{44}

In this article, we utilise the cooperative-string model^{31} – recently developed and successfully applied to thin glassy films – for the present case of glassy nanoparticles. The general philosophy of our approach consists of combining classical free-volume and cooperativity arguments to the more recent observations of a specific string-like character of the cooperatively rearranging regions, within a minimal kinetic model allowing us to address analytically and quantitatively the confinement-induced and interfacial effects. After recalling the main ingredients of the bulk description, we turn to its modification in spherical geometry and discuss the implications for experiments. In particular, we characterize the extent of the mobile-layer region as a function of temperature, and we describe the glass-transition temperature reductions as a function of particle radius, providing predictions for experiments, including the existence of a minimal radius below which vitrification never occurs. One purpose of this work is that it allows a natural separation between the effects intrinsic to glass formation, and those related to the polymeric nature of the materials for which even the thin-film geometry^{12} has so far defied a proper theoretical description, despite promising ideas.^{45,46}

As a preliminary, we would like to stress an important point. For the sake of simplicity, we map the real liquid state to a simple hard-sphere liquid and thus fully neglect the enthalpic contributions in the activation barriers for relaxation. Stated differently, we assume that entropic effects are dominant in the critical slowing down of supercooled liquids. This is also what the Gibbs–DiMarzio approach to glass formation in polymers would suggest, with the underlying transition essentially determined by a vanishing of the configurational entropy.^{48} In fact, we are not trying to come up with a definitive and detailed theory of glassy dynamics, but rather address the much simpler question: what physics are we able to get out of the ideas of caging and cooperative motion? In particular, more than the bulk description already developed by Adam and Gibbs^{49} or Wolynes,^{58} our goal is to understand how such a minimal free-volume model will exhibit finite-size effects that are not always easily determined in other approaches.

Let us consider a dense assembly of small molecules with size λ_{V} and average intermolecular distance λ. The volume fraction is thus ϕ ∝ (λ_{V}/λ)^{3}. A test molecule sits in a cage of volume ∼λ^{3}, with gates of length L ∼ λ − λ_{V}. We set that a typical non-cooperative liquid-like local relaxation requires L > L_{c} = λ_{c} − λ_{V}, or equivalently λ to be larger than λ_{c} – the so-called onset of cooperativity, with volume fraction ϕ_{c}. Also, when λ ∼ λ_{V}, the gates are completely closed (L ∼ 0) and the system is at kinetic arrest, with volume fraction ϕ_{V}. For λ_{V} < λ < λ_{c}, the relaxation is possible but necessarily collective. It requires a random string-like cooperative motion involving at least N* − 1 neighbours of the test molecule, that provide a total space (N* − 1) L ∼ L_{c} − L by getting in close contact with each other. The test molecule thus sees a temporary larger gate, of length L_{c}, and can exit the cage. Therefore, one gets the scaling expression of the minimal number of molecules needed for a local relaxation, i.e. the so-called cooperativity:

(1) |

By introducing the necessity of coherence between molecular motions within a cooperative rearrangement, one showed^{31} that in a typical cooperative string made of N* molecules the relaxation rate τ^{−1} follows:

(2) |

Since, in the temperature range considered, the thermal expansion coefficient α = −(1/ϕ)dϕ/dT of the supercooled liquid is almost constant, one has:

ϕ(T) = ϕ_{V}[1 − α(T − T_{V})], | (3) |

(4) |

As a first remark, τ_{c}/τ_{0} should be a constant only for hard spheres. For a real liquid, the latter dimensionless relaxation time is rather expected to follow an Arrhenius law τ_{c}/τ_{0} = exp(T_{a}/T), where k_{B}T_{a} is an activation energy barrier proportional to the cohesive interaction strength, and k_{B} is the Boltzmann constant. In that case, eqn (4) would be replaced by the leading-order expression near the kinetic-arrest point:

(5) |

As the bulk relaxation process presented here consists of random cooperative strings involving N* molecules, one can minimally describe them through ideal random walks. The length scale ξ of the cooperatively rearranging regions is thus of the form , near the kinetic arrest point. Note that if we rather use more realistic self-avoiding random walks, the exponent becomes ∼0.6 instead of 1/2 – but such a refinement would be at the cost of mathematical simplicity for the confinement effects discussed below. Invoking eqn (1) and (3), one obtains the temperature-dependent expression of the cooperativity:

(6) |

(7) |

Fig. 1 Two string-like cooperative paths in a supercooled spherical nanoparticle of radius R. Relaxation of a test molecule (green) at a distance r from the sphere center can occur through either a bulk cooperative string (blue) of size ξ (eqn (7)), or a truncated string (red) touching the interface. |

Because our minimal description of the local relaxation process relies on random cooperative strings involving N* molecules, we can use a first-passage argument in the limit of large N*, in order to determine N_{s}*. Before doing so, we make the following remark. The typical string length in numerical simulations is rather short.^{59} This is related to the fact that, due to computational time constraints, those simulations were performed at relatively high temperatures – basically near the caging onset – and thus address the onset of cooperative motion. In contrast, in experiments, such as the ones with vibrated granular beads,^{62} or repulsive colloids^{63} for instance, notably longer structures are seen. The clear advantage of working near the kinetic-arrest point is to get a Brownian description, and thus tractable analytical results using first-passage probability densities. This is an idealized asymptotic view valid only near the divergence point. But, as for critical phenomena in continuous phase transitions, it may allow to extract some universal features that might still be relevant away from the divergence point.

For that purpose, we define n_{0} as the number of molecular units at which a given realization of a random string reaches the interface for the first “time”. If n_{0} ≥ N*, the string is bulk-like; if n_{0} < N*, the string is truncated by the interface. Therefore, the important quantity here is the density of probability g(t) of the first-passage “time” t = n_{0}/N* at the interface, located at dimensionless radial position R/ξ, of a 3D Brownian process starting at dimensionless radial position r/ξ, with r < R. Below, we briefly summarise the main mathematical steps allowing to obtain g(t) explicitly. The Laplace transform of g(t) can be written as:^{66,67}

(8) |

(9) |

(10) |

(11) |

(12) |

Knowing the first-passage probability density g(t), one can now compute the average local cooperativity N_{s}* by averaging the minimum between N* and n_{0}:

N_{s}*(r, R, T) = N*(T)〈min(1,t)〉_{t} | (13) |

(14) |

(15) |

(16) |

(17) |

(18) |

Fig. 2 Predicted surface mobile-layer thicknesses h_{m} of spherical polystyrene nanoparticles as a function of temperature T, according to eqn (16) and (17), for different sphere radii as indicated. We used the bulk glass-transition temperature T^{bulk}_{g} = 371 K,^{69} and the onset temperature T_{c} = 463 K.^{57,70} We fixed the molecular size λ_{V} = 3.7 nm, and the Vogel temperature T_{V} = 322 K, to the values previously obtained for the thin-film geometry.^{31} Note that we replaced the +∞ bound by 25 in eqn (16), and checked that it provides sufficiently precise numerical estimates. For comparison, the dashed line indicates the flat-interface result used for the thin-film geometry.^{31} |

We can now determine the effective glass-transition temperature measured in nanoparticles of radius R, by using the following criterion:^{26,28} the transition occurs when half of the sample volume is liquid and the other half is glassy, i.e. (R − h_{m})/R = 2^{−1/3} for a sphere. Introducing , and using eqn (17), we get:

(19) |

Fig. 3 Comparison between experimental data (symbols) for the reduced glass-transition temperature of spherical polystyrene nanoparticles^{35,37,40} of radius R, and the theory (line) given by eqn (19) – that invokes eqn (16) through . The fixed parameters are the bulk glass-transition temperature T^{bulk}_{g} = 371 K,^{69} and the onset temperature T_{c} = 463 K.^{57,70} The two adjustable parameters are the molecular size λ_{V} = 3.7 nm, and the Vogel temperature T_{V} = 322 K, that were fixed to the values previously obtained for the thin-film geometry.^{31} Note that we replaced the +∞ bound by 25 in eqn (16), and checked that it provides sufficiently precise numerical estimates. |

At this point, it is important to make a few remarks. First, if we shift vertically the data of Zhang et al.,^{37} in order to enforce equal T^{bulk}_{g} values between all studies, the general agreement to literature data becomes comparable to the thin-film case.^{31} Secondly, the presence of residual surfactants – a necessary ingredient in many nanosphere preparation techniques – has been shown to have a large effect on the magnitude of the glass-transition temperature reductions in thin films,^{43} and could thus be responsible for part of the experimental scatter. Thirdly, spherical polymer nanoparticles may well be like freestanding films, in that no part of the sample is on a substrate. It is thus fruitful here to remember the case of freestanding polystyrene thin films.^{12} In those samples, two different types of behaviour were observed: (i) for low molecular weights, there are reductions in the glass-transition temperature that do not depend on the molecular weight, and which are similar to the ones in supported films – well captured by our cooperative-string model;^{31} (ii) in contrast, for large molecular weights, the reductions in the glass-transition temperature are much more pronounced, and exhibit a dependence with molecular weight, suggesting another – polymeric – relaxation mode,^{45,46} still to be described theoretically in quantitative detail.

To conclude, in view of the robustness of our cooperative-string model for bulk samples and supported films,^{31} and given the present results and discussion, this work could be set as a theoretical framework for describing the glassy dynamics in surfactant-free monodisperse low-molecular-weight spherical nanoparticles. Beyond the radius-dependent reduction in the glass-transition temperature, the model offers a preliminary prediction on the surface mobile-layer thickness as a function of temperature, and sets the existence of a minimal sphere radius, below which vitrification never occurs. As such, our results reveal important constraints on potential applications, and may serve as a guiding tool for future fundamental studies around the glass transition, in confinement, and at interfaces.

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