q lm1 q lm2 q lm3 (1) m 1,m 2,m 3,m 1 +m 2 +m 3 =0 m 1 m 2 m 3 l l l

Transcription

1 SUPPLEMENTARY INFORMATION Bond-orientational order parameters. We use a particle-level bond-orientational order parameter defined as follows. where the coefficients W l l l l q lm1 q lm2 q lm3 (1) m 1,m 2,m 3,m 1 +m 2 +m 3 =0 m 1 m 2 m 3 l l l m 1 m 2 m 3 are the Wigner 3J symbols and the local order around each particle is characterised by q lm 1 N b (i) Y lm (r ij ), N b (i) j=1 where Y lm are spherical harmonics [16]. W 6 is then calculated for each particle, provided it has three or more neighbours. The W 6 distributions presented are then weighted by the number of neighbours. Further Experimental details. The data were collected on a Leica SP5 confocal microscope, fitted with a resonant scanner. We tracked the co-ordinates of each particle with a precision of around 100 nm [22,27]. We investigated the consequences of the co-ordinate tracking errors and colloid polydispersity, with Monte-Carlo simulation. Very little change in the cluster populations was found, and we conclude that our main results are insensitive to these effects. The coordinates were tracking time using methods similar to [32]. The MSD presented in Figs. 1f and 4c were calculated from xy data only [32]. Brownian Dynamics Simulations. To reproduce the experiments as closely as was practical, we carried out standard Brownian dynamics simulations according to the Morse potential (Supplementary Fig. 2). βu M (r) = ɛe ρ0(1 r/σ) (e ρ0(1 r/σ) 2) (2) with the range parameter zρ 0 = 30. We used typically 1000 particles and a timestep of 10 5 τ B. We matched the experimental polydispersity with particle diameters distributed 1

2 according to a Gaussian of standard deviation 0.04σ, and choose an identical bond length of 0.18σ. The simulations were carried out in the overdamped limit. Periodic boundary conditions as employed for the simulations themselves were used in the TCC analysis. Non-ergodic state points were aged for 10τ B prior to analysis, a comparable time to the experiments. Following this aging, no change was detected, either in the simulations, nor in the experiments. Mean squared displacements are shown in Supplementary Fig. 4, and the LFS population in Supplementary Fig. 5. The MSD reveals a transition to slow dynamics between ε = 3.0 and ε = 4.0, allowing us to define ε/ε G in much the same way as for the experiments, except that now the interaction well depth is parameterised by ɛ, (Eq. 2), rather than by c P. Supplementary Fig. 6 shows the LFS lifetime, in terms of the particle residency within LFS, as defined by < c LFS (0)c LFS (t) >. This clearly diverges upon gelation. Ih icosahedron C3v D5h D3h C2v C2v Cs m D5h pentagonal bipyramid D3h triangular dipyramid Oh octahedron Supplementary Fig. 1: W 6 of the various LFS investigated. Point group symmetry notation follows [25]. Colours denote LFS size m. The icosahedron (purple triangle) is not considered to be an LFS of this system, as m = 13 D5 h has a lower potential energy [25]. By no means all the LFS have a strongly negative W 6. 2

3 4 u(r)/k B T depletion Morse 0 =30 Lennard-Jones r/ Supplementary Fig. 2: Comparison of the model interactions. The depletion interaction, which describes this system [27], is plotted as the red line. This is compared with the Morse interaction, which was used to calculated the potential energy minima of the isolated clusters [25]. The Morse interaction is plotted for the range parameter ρ 0 = 30 (green line). We assumed that the LFS for both the short-ranged Morse and depletion interactions were identical. Also shown is the Lennard-Jones 6-12 interaction (blue line). 3

4 6 g(r) r/ Supplementary Fig. 3: Radial distribution functions. The values on the right correspond to c P /c PG. Successive distributions have been shifted up the ordinate axis by an increment of unity for clarity. <r.r>/ t/ B Supplementary Fig. 4: Mean squared displacement from Brownian Dynamics simulations. The legend denotes the reduced temperature ɛ/ɛ G as defined in in the text. Non-ergodic data (ɛ/ɛ G 1) are aged for 100τ B. The dashed black line has a slope of unity, indicating diffusive behaviour. We identify the orders of magnitude reduction in MSD 4

5 between ɛ/ɛ G = 0.86 and 1.14 (ɛ=3.0 and 4.0 respectively, Eq. 2) with the onset of slow dynamics and non-ergodicity. ergodic liquid dynamically arrested total FCC Supplementary Fig. 5: Proportion of particles in LFS as a function of the reduced temperature ɛ/ɛ G. Note very similar behaviour to Fig <c LFS (0)c LFS (t)> t/ B Supplementary Fig. 6: Particle residency in LFS < c LFS (0)c LFS (t) > from Brownian Dynamics simulations. Legend denotes the reduced temperature ɛ/ɛ G as defined in 5

6 in the text. In the BD simulations, the accurate knowledge of the particle coordinates leads to a divergence in LFS lifetime upon arrest. 6