Effect of Cooling Rate on the Microstructure of Colloidal Glass
2017 Volume 58 Issue 3 Pages 420-422
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2017 Volume 58 Issue 3 Pages 420-422
The cooling rate with which the liquid is cooled has tremendous impact on the macroscopic properties of amorphous solids, but little information on the underlying mechanism for this dependence is available, mainly due to the lack of clear characterization on the microstructural variation induced by cooling rate. We built a colloidal glass to obtain its direct three-dimensional configuration by using laser scanning confocal microscopy and investigate the effect of cooling rate on microstructure. By quantifying coordination numbers and bond-angle distribution, we give evidence that the icosahedral-like structure is the most frequent local structure and more favored by the lower cooling rate.
An amorphous solid is normally formed by supercooling a viscous liquid fast enough to avoid crystallization1). The transition from a liquid state to an amorphous solid state is essentially kinetic, resulting from a “falling out of equilibrium” of the liquid as the structural relaxation time increases with decreasing temperature and exceeds the time scale of the experiment2). Thus, the resulting amorphous solid is not in thermal equilibrium and its properties strongly depend on the thermal history, such as the cooling rate with which the liquid is cooled3). Such dependencies exist indeed and have been shown in various experiments and computer simulations, in which the thermal, dynamic, and mechanical properties of amorphous solids are found to be sensitive to their cooling rates4–7). Although much has already been known about the effects of cooling rate on the macroscopic properties of amorphous solids, little information on the underlying mechanism for this link is available. This is mainly due to the lack of clear characterization on the microstructural variation induced by different cooling rates, since the properties of amorphous solids are essentially related to their microstructures8). Thus, figuring out the difference in microstructures of amorphous solids produced with different cooling rates is the key to understand the dependencies of properties on the cooling rate. However, it is prohibitively difficult to directly investigate this within atomic and molecular solids.
By contrast, colloidal glass of micrometer-sized spherical particles can serve as an excellent model to investigate how cooling rate influences the microstructures as the larger size of colloidal particles makes them much more experimentally accessible9–11). The colloidal particles can be directly observed in real time and their positions in three dimensions can be determined accurately by high-speed confocal microscopy. Subsequent image analysis enables us to calculate the structural properties, providing an accurate picture of interrelationship between cooling rate and microstructures. Thus, in the present work, we built a colloidal glass to obtain its direct three-dimensional configuration by using laser scanning confocal microscopy. By quantifying coordination numbers and bond-angle distribution, we found that the icosahedral-like structure is the most frequent local structure and more favored by the lower cooling rate.
We used 1.55-μm diameter colloidal silica particles with a polydispersity smaller than 3.5% to prepare a colloidal system containing more than 4 × 109 colloidal particles. The silica particles were suspended in a mixture of deionized water and dimethyl sulfoxide. To make the particles appear as dark spots on a bright background under fluorescence microscopy, we dyed them with fluorescein-NaOH solution. The schematic of the sample cell is shown in Fig. 1. Due to the density difference between colloidal particles and their surrounding fluid, the colloidal structures were constructed by sedimentation under gravity. By using a high-speed confocal microscope, we acquired three-dimensional scans of our sample yielding a 77 × 77 × 23 μm3 observation volume for each image stack. Each image stack takes 150 s. We identified particle positions in 3D with a horizontal accuracy of 0.03 μm and a vertical accuracy of 0.05 μm12).
Schematic showing the experimental set-up with colloidal glass.
In hard-sphere system, the viscosity approaching the glass transition varies with volume fraction and can be described as $\eta = \eta_0 \ {\rm exp} [ \nu \phi /( \phi_0 - \phi )]$13). Correspondingly, the viscosity of a molecular liquid approaching the glass transition that varies with temperature (T) can be described by Vogel-Fulcher-Tammann (VFT) equation, $\eta = \eta_0 \ {\rm exp} [DT_0 /(T - T_0 )]$14). The above two equations are similar in form, suggesting that $\phi $ plays a similar role in hard-sphere system as T in the usual liquids15). In other words, the control parameter of hard-sphere system is $\phi $ rather than T, and the effective temperature of hard-sphere system is $ T_{eff} = 1/ \phi $16). Then the effective cooling rate ($\dot T_{\mathit{eff}}$) of the hard-sphere system can be expressed as
| \[\dot T_{\mathit{eff}} = \frac{1/ \phi_1 - 1/ \phi_2}{\Delta t}\] | (1) |
As described above, we built a colloidal glass by sedimentation under gravity. There may be differences in $ \dot T_{\mathit{eff}}$ along the height of the colloidal glass. To check this assumption, we directly viewed the formation process of the colloidal glass and calculated the $ \dot T_{\mathit{eff}}$ at different heights of the colloidal glass. Figure 2 (a) to (d) are four reconstruct colloidal structures in 3-μm-thick x-z section centered at y = 10 μm at t = 0 s, 150 s, 450 s and 1650 s, respectively. The glass formation process can be seen clearly in this figure. Along the z direction, the packing densities of the colloidal system vary a lot at the initial time (t = 0 s), but no apparent difference can be detected after 1650 s, strongly suggesting that $ \dot T_{\mathit{eff}}$ is different with the height of colloidal glass. Clearly, according to eq. (1), the $ \dot T_{\mathit{eff}}$ gradually increase with the height increasing. This gradual variation of $ \dot T_{\mathit{eff}}$ along the height of colloidal glass offers us a good chance to study the effect of cooling rate on the microstructures.
Four snapshots during sedimentation of a colloidal glass. (a) to (d) are 3-μm-thick x-z sections centered at y = 10 μm at time (a) t = 0 s, (b) t = 150 s, (c) t = 450 s, and (d) t = 1650 s.
To clarify the role of cooling rate in the structure of colloidal glass, we investigate the dependence of coordination numbers on $ \dot T_{\mathit{eff}}$. We focus on the microstructure of the colloidal system at t = 1650 s, when the stable colloidal glass just forms through verification and its microstructure may be affected by $ \dot T_{\mathit{eff}}$ more evidently. We define the coordination number (N) of a particle to be the number of particles that are closer to the first particle than rmin. Here, rmin is the location of the first minimum in radial distribution function (RDF). Figure 3(a) shows the typical distribution P(N) of the colloidal glass in 3-μm-thick x-y section centered at z = 2 μm. We recognize that the most frequent value of the coordination numbers for the colloidal glass is 12. This indicates that more icosahedral structure may be formed in the colloidal glass, since particles of close-packed system tend to pack locally in an icosahedral structure with a coordination number of 12, which is the most compact and energetically stable structure17).
(a) The typical distribution of coordination number P(N) of the colloidal glass (t = 1650 s) in 3-μm-thick x-y section centered at z = 2 μm. (b) The bond angle distribution for the particles with the coordination number of 12 in the colloidal glass (t = 1650 s). (c) The distribution of coordination number P(N) of the colloidal glass (t = 1650 s) in 3-μm-thick x-y section centered at z = 2 μm, 5 μm, 8 μm, 11 μm, 14 μm, 17 μm, and 20 μm. The arrow indicates the increasing of the centres of 3-μm-thick x-y sections in colloidal glass.
To validate whether the nature of this arrangement is really icosahedral structure, we calculate the bond angle distribution of the colloidal glass. Note that the icosahedral-type packing should produce peaks of bond angle distribution at the angles 63.4° and 116.6° 18,19). We define a “bond” as the line connecting two neighboring particles. Then the bond angle is considered as the angle between two bonds that connected to the same particle. Figure 3(b) shows the bond angle distribution for the particles with the coordination number of 12 in the colloidal glass at t = 1650 s. Dashed lines shown in this figure correspond to the perfect icosahedral angles of 63.4° and 116.6°. It can be seen form this figure that the distribution curve indeed presents two peaks. However, the two peaks locate at 59.4° and 114.7° and are broad, indicating that the microstructure is not perfect icosahedral structures, but instead an icosahedral-like structure. Though the structure with the coordination number of 12 is not a perfect icosahedral, its icosahedral-like packing is still much dense and energetically competitive20). Figure 3(c) shows the distribution P(N) of the colloidal glass (t = 1650 s) in 3-μm-thick x-y section centered at z = 2 μm, 5 μm, 8 μm, 11 μm, 14 μm, 17 μm, and 20 μm, respectively. The overall distribution P(N) of the colloidal glass at different heights displays only a small dependence on the height of colloidal glass. The most pronounced change is that the probability of the coordination number 12 shows a tendency to decrease to smaller values with increasing height, demonstrating that the formation of icosahedral-like structure is more favored by lower cooling rate. This is because the icosahedral-like structure is lower in energy than the amorphous structure. Thus for a lower cooling rate, the undercooled melt has more time to develop icosahedral-like structure to reduce the energy of the system.
We have built a colloidal glass to investigate the dependence of the microstructure on the cooling rate. By investigating coordination numbers and bond-angle distribution, we give evidence that the icosahedral-like structure is the most frequent local structure and more favored by the lower cooling rate.
This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51401041 and 51671042, China Postdoctoral Science Foundation under Grant No. 2015M570242, and Basic Research Project of Key Laboratory of Education Department of Liaoning Province under Grant No. LZ2015011.
