Effect of Cooling Rate on the Microstructure of Colloidal Glass

Yongzhe Wang1, Lidan Qu2, Hongge Li1, Hao Zhang1, Yilin Wang1 and Yunzhuo Lu1,*

1School of Materials Science and Engineering, Dalian Jiaotong University, Dalian 116028, People’s Republic of China2Handling Machinery Manufacturing Deparment, Dalian Huarui Heavy Industry Group Co., Ltd., Dalian 116013, People’s Republic of China

The cooling rate with which the liquid is cooled has tremendous impact on the macroscopic properties of amorphous solids, but little in-formation 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 con�guration 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. [doi:10.2320/matertrans.M2016408]

(Received November 14, 2016; Accepted December 14, 2016; Published February 25, 2017)

Keywords:　 amorphous materials, colloidal glass, cooling rate, microstructure

1.　 Introduction

An amorphous solid is normally formed by supercooling a viscous liquid fast enough to avoid crystallization1). The tran-sition from a liquid state to an amorphous solid state is essen-tially kinetic, resulting from a “falling out of equilibrium” of the liquid as the structural relaxation time increases with de-creasing temperature and exceeds the time scale of the exper-iment2). 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 amor-phous 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 sol-ids are essentially related to their microstructures8). Thus, �g-uring out the difference in microstructures of amorphous sol-ids produced with different cooling rates is the key to under-stand the dependencies of properties on the cooling rate. However, it is prohibitively dif�cult 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 in�uences the microstructures as the larger size of colloidal particles makes them much more experimentally accessible9–11). The colloidal particles can be directly ob-served in real time and their positions in three dimensions can be determined accurately by high-speed confocal microsco-py. Subsequent image analysis enables us to calculate the structural properties, providing an accurate picture of interre-lationship between cooling rate and microstructures. Thus, in the present work, we built a colloidal glass to obtain its direct three-dimensional con�guration by using laser scanning con-

focal microscopy. By quantifying coordination numbers and bond-angle distribution, we found that the icosahedral-like structure is the most frequent local structure and more fa-vored by the lower cooling rate.

2.　 Experiments

We used 1.55-μm diameter colloidal silica particles with a polydispersity smaller than 3.5% to prepare a colloidal sys-tem containing more than 4 ×  109 colloidal particles. The sil-ica 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 �uorescence microscopy, we dyed them with �uorescein-NaOH solution. The schemat-ic of the sample cell is shown in Fig. 1. Due to the density difference between colloidal particles and their surrounding �uid, the colloidal structures were constructed by sedimenta-tion under gravity. By using a high-speed confocal micro-scope, we acquired three-dimensional scans of our sample yielding a 77 ×  77 ×  23 μm3 observation volume for each im-age stack. Each image stack takes 150 s. We identi�ed parti-cle positions in 3D with a horizontal accuracy of 0.03 μm and a vertical accuracy of 0.05 μm12).

* Corresponding author, E-mail: yunzhuohit@gmail.com Fig. 1　Schematic showing the experimental set-up with colloidal glass.

Materials Transactions, Vol. 58, No. 3 (2017) pp. 420 to 422 ©2017 The Japan Institute of Metals and Materials

3.　 Results and Discussion

In hard-sphere system, the viscosity approaching the glass transition varies with volume fraction and can be described as η =  η0 exp[νϕ/(ϕ0 −  ϕ)]13). Correspondingly, the viscosity of a molecular liquid approaching the glass transition that varies with temperature (T) can be described by Vogel-Fulcher-Tam-mann (VFT) equation, η =  η0 exp[DT0/(T −  T0)]14). The above two equations are similar in form, suggesting that ϕ 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 ϕ rather than T, and the effective temperature of hard-sphere system is Teff =  1/ϕ16). Then the effective cooling rate (Teff  ) of the hard-sphere system can be expressed as

Teff =1/φ1 − 1/φ2

∆t (1)

where ϕ1 and ϕ2 are the volume fractions of colloidal system at time t1 and t2, respectively. Δt =  t2 −  t1 is the time interval between t1 and t2.

As described above, we built a colloidal glass by sedimen-tation under gravity. There may be differences in Teff   along the height of the colloidal glass. To check this assumption, we directly viewed the formation process of the colloidal glass and calculated the Teff   at different heights of the colloidal glass. Figure 2 (a) to (d) are four reconstruct colloidal struc-tures 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 �gure. Along the z direc-tion, 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 Teff   is different

with the height of colloidal glass. Clearly, according to eq. (1), the Teff   gradually increase with the height increasing. This gradual variation of Teff   along the height of colloidal glass offers us a good chance to study the effect of cooling rate on the microstructures.

To clarify the role of cooling rate in the structure of colloi-dal glass, we investigate the dependence of coordination numbers on Teff  . We focus on the microstructure of the colloi-dal system at t =  1650 s, when the stable colloidal glass just forms through veri�cation and its microstructure may be af-fected by Teff   more evidently. We de�ne the coordination number (N) of a particle to be the number of particles that are closer to the �rst particle than rmin. Here, rmin is the location of the �rst minimum in radial distribution function (RDF). Figure 3(a) shows the typical distribution P(N) of the colloi-dal 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 struc-ture17).

To validate whether the nature of this arrangement is really icosahedral structure, we calculate the bond angle distribu-tion 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 de�ne a “bond” as the

Fig. 2　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.

Fig. 3　(a) The typical distribution of coordination number P(N) of the col-loidal 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 coordina-tion 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.

421Effect of Cooling Rate on the Microstructure of Colloidal Glass

line connecting two neighboring particles. Then the bond an-gle is considered as the angle between two bonds that con-nected 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 �gure correspond to the perfect icosahedral angles of 63.4° and 116.6°. It can be seen form this �gure that the dis-tribution 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 icosahe-dral, its icosahedral-like packing is still much dense and ener-getically competitive20). Figure 3(c) shows the distribution P(N) of the colloidal glass (t =  1650 s) in 3-μm-thick x-y sec-tion 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 de-pendence on the height of colloidal glass. The most pro-nounced 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 cool-ing rate, the undercooled melt has more time to develop ico-sahedral-like structure to reduce the energy of the system.

4.　 Conclusion

We have built a colloidal glass to investigate the depen-dence of the microstructure on the cooling rate. By investigat-ing 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 cool-ing rate.

Acknowledgements

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.

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