pubs.acs.org/Langmuir © 2009 American Chemical Society
Contact between Submicrometer Silica Spheres Xiao-Dong Wang, Zheng-Xiang Shen, Jin-Long Zhang, Hong-Fei Jiao, Xin-Bin Cheng, Xiao-Wen Ye, Ling-Yan Chen, and Zhan-Shan Wang* Institute of Precision Optical Engineering, Department of Physics, Tongji University, Shanghai 200092, China Received October 12, 2009. Revised Manuscript Received November 27, 2009 Recently, the scope of the investigation of the deformation mechanism extended to the micrometer and submicrometer regimes. The sphere-substrate contact method was usually used because it is rather difficult to make two micrometer or submicrometer spheres contact each other precisely. Here, we used the sphere-sphere contact method via a novel, simple process to investigate the deformation of spheres. The silica particle size ranges from 400 to 900 nm. Traditionally, the harder the particle, the smaller both the contact radius and the adhesion force. Therefore, it is widely accepted that silica particles should undergo elastic deformation, but we found that silica particles underwent plastic deformation rather than elastic deformation because of van der Waals interaction. The contact radii were observed by scanning electron microscopy (SEM).
Introduction The deformation behavior of particles plays an important role in many areas such as semiconductor technology and drug delivery.1-3 In the past 40 years, much work has been done to study the mechanism of particle deformation resulting from adhesive contact.4-7 Several models were proposed on this subject. In 1971, Johnson, Kendall, and Roberts4 (hereafter referred to as JKR) revealed that the contact radius varied as the particle radius to the 2/3 power in the absence of an externally applied load. Then, Derjaguin, Muller, and Toporov5 (hereafter referred to as DMT) showed the 2/3 power-law dependence of the contact radius on the particle radius under zero load. They calculated the contact radii based on the molecular interaction. It was found that the deformation of the particles was caused by the repulsive components of the interaction forces. Both the JKR and DMT models assumed that the contact radii were small and only elastic deformation occurred. However, the values of the contact radii predicted by the DMT model were approximately half of those based on the JKR theory. Muller, Yushchenko, and Derjaguin6 (hereafter referred to as MYD) later introduced a general theory in which the JKR and DMT models were valid in different cases. They indicated that for the case of large particles, higher surface energy, and lower Young’s modulus the JKR model is applied; for the case of small particles, lower surface energy, and higher Young’s modulus, the DMT model is valid. In 1984, Maugis and Pollock7 (hereafter referred to as MP) extended the JKR model to include the plastic deformation. They proposed that the power-law *Corresponding author. Tel: þ86-21-65984652. Fax: þ86-21-65984652. E-mail:
[email protected]. (1) Kuo-Kang Liu, J. J. Phys. D: Appl. Phys. 2006, 39, R189–R199. (2) Barthel, E. J. Phys. D: Appl. Phys. 2008, 41, 163001. (3) Eichenlaub, S.; Kumar, G.; Beaudoin, S. J. Colloid Interface Sci. 2006, 299, 656–664. (4) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (5) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314. (6) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 77, 91. (7) Maugis, D.; Pollock, H. M. Acta Metal. 1984, 32, 1323.
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dependence of the contact radius on the particle radius was 0.5 for the case of plastic deformation. During research on the deformation mechanism of spheres, two methods were usually utilized: (1) sphere-sphere contact and (2) sphere-substrate contact. In earlier studies, researchers mainly focused on the deformation of the millimeter or centimeter spheres. The sphere-sphere contact method was mainly used. In 1896, Hertz investigated the contact between two smooth elastic spheres. He found that both the size and shape of the zone of contact resulted from the elastic deformation of the spheres.8 Roberts, Kendall, and Johnson also studied the deformation mechanism between spheres.9-11 They used centimeter spheres in their experiments. They found that contact areas between spheres were considerably larger than those predicted by Hertz at low loads and tended toward a constant finite value as the load was reduced to zero. After that, the scope of the investigation of the deformation mechanism extended to the micrometer and submicrometer regimes. The sphere-substrate contact method was usually used because it is rather difficult to make two micrometer or submicrometer spheres contact each other precisely.12-15 Unfortunately, the sphere and the substrate usually made from different materials introduced some error into the calculation, and their different geometrical shapes also introduced some error in the measurement. Thus, it is necessary and meaningful to find a way to make two micrometer or submicrometer spheres contact each other. In this article, a novel, simple process was designed to make two submicrometer silica spheres contact each other. The silica particle size ranges from 400 to 900 nm. The particles generated the deformation as a result of attractive forces. The deformation mechanism was discussed. We hope that our work will give the old sphere-sphere contact method new life. (8) Hertz, H. Miscellaneous Papers; : Macmillan: London, 1896; pp 146-183. (9) Roberts, A. D. Ph.D. Dissertation, Cambridge University: England, 1968. (10) Kendal, K. Ph.D. Dissertation, Cambridge University: England, 1969. (11) Johnson, K. L. Br. J. Appl. Phys. 1958, 9, 199. (12) Rimai, D. S.; DeMejo, L. P.; Bowen, R. C. J. Appl. Phys. 1989, 66, 3574. (13) Rimai, D. S.; Quesnel, D. J.; Bowen, R. C. Langmuir 2001, 17, 6946–6952. (14) Rimai, D. S.; Schaefer, D. M.; Bowen, R. C.; Quesnel, D. J. Langmuir 2002, 18, 4592–4597. (15) Dejesus, M. C.; Rimai, D. S.; Quesnel, D. J. Langmuir 2006, 22, 729–735.
Published on Web 12/11/2009
DOI: 10.1021/la9038446
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Experiment Silica particles in the size range of 400-900 nm were prepared by the traditional St€ ober process;16,17 tetraethyl-orthosilicate, ammonia, water, and alcohol are chemical reaction materials.17 Si-O-Si and O-H are the main chemical bonds of silica spheres.17 The experimental process is listed below: (1) Silica particles were suspended in the alcohol. The alcohol was distilled prior to use. (2) A drop of the suspension (approximately 0.2 mL) was diluted in 30 mL of distilled alcohol. (3) A drop of diluted suspension (approximately 0.2 mL) taken out of the 30 mL alcohol solution was dropped onto the center of the fused silica substrate. Then, the alcohol began to spread gently on the substrate. The spheres had a chance to contact each other during the spreading process. Finally, only spheres were left on the substrate after the alcohol evaporated. The concentration of silica spheres in alcohol was approximately 0.3 (solid content). The substrate was carefully cleaned before it was used, and it is observed by darkfield microscope that there were only 0-5 particles per about 0.3 mm2 after it was cleaned; the contact angle between the alcohol and the cleaned substrate is zero. (4) The samples were kept in a cabinet at 25 °C and 45% RH for 2 weeks in order to ensure that some contacting spheres reached the force balance. There are two key factors that determine whether our experiment will succeed; one is the concentration of silica spheres in the 30 mL alcohol suspension, and the other is the cleanness of the particles, the substrate, and the alcohol. Multilayer particles will be formed if the concentration of silica spheres in the 30 mL alcohol suspension is too high. This may introduce an extra applied load onto the particles and lead to some uncertainty and difficulty in the followed measurement and analysis. Thus, it is preferred that monolayer particles be dispersed on the substrate. Moreover, we assumed that the impact force of the moving sphere acting on the other sphere was small and can be ignored because the spread velocity of the alcohol on the substrate was small. The contact radii of the silica particles were determined with a field emission scanning electron microscope (FE-SEM, JEOL JFC-1600, Japan). To eliminate the charging effect for the insulating material, the samples were coated with a 9.2-nm-thick layer of Pt by sputtering with argon at 2.5 kV, 10 mA for 60 s.
Results and Discussion Figure 1 shows the contact between silica spheres with similar sizes. Silica particles in the size range of 400-900 nm generated the deformation. The reduced radius ranges from 93.7 to 232.4 nm. The reduced radius is obtained by 1 1 1 ¼ þ Rreduced R1 R2 The sphericity of particles has a great influence on the correctness of this research. Table 1 demonstrates the sphericity of each sphere in terms of the maximum deviation from the average diameter. As shown in Table 1, the maximum deviation from the average diameter is less than 1.5%, which implies that the particles have excellent sphericity. Rimai’s group did extensive work on the adhesion of particles to substrates due to surface forces.12-15 It was found that the contact radius varied as the particle radius to an anomalous (16) St€ober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62–69. (17) Wang, X. D.; Shen, Z. X.; Sang, T.; Cheng, X. B.; Li, M. F.; Chen, L. Y.; Wang, Z. S. J. Colloid Interface Sci. 2010, 341, 23–29.
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Figure 1. Contact between silica spheres with similar sizes. The two silica particles with reduced radii of (a) 93.7, (b) 104.9, (c) 119.9, (d) 122.9, (e) 171.2, (f) 180.8, (g) 208.7, (h) 212.4, and (i) 232.4 nm generated the deformation.
power when the particle was in contact with compliant substrates; furthermore, over time, the particle appears to become engulfed by the substrate. They successfully designed an elaborate experimental process and a perfect discussion mode on this subject. Here, we follow their research mode to discuss our results. First, a power-law dependence is obtained through a log-log plot between the contact radius a and the reduced particle radius R. According to the obtained power-law dependence, a certain theory (JKR, DMT, or MP) that holds for the experimental results could be determined. Second, the experimental value of the work of adhesion could be calculated through the slope of the line plot between contact radius a and R0.5 (MP theory as an example), and one can verify the correctness of the determined theory by comparing this experimental value of the work of adhesion with the theoretical value. Finally, one can judge whether the adhesion Langmuir 2010, 26(8), 5583–5586
Wang et al.
Article Table 1. Sphericicity of Spheres Used in the Experiment d1 (nm)
no. 1 2 3 4 5 6 7 8 9
d2 (nm)
1
2
3
d1 average (nm)
maximum deviation
1
2
3
d2 average (nm)
maximum deviation
394.7 441.5 486.5 516.4 692.0 731.1 847.2 875.2 936.9
396.7 440.1 489.1 512.3 699 738.6 850.3 877.7 933.2
391.5 431.0 492.6 515.8 697.4 737.8 844.1 874.3 934.5
394.3 437.5 489.4 514.8 696.1 735.8 847.2 875.7 934.8
0.71% -1.49% 0.60% -0.49% 0.59% -0.64% 0.37% 0.22% 0.22%
392.2 441.4 507.8 510.2 717.7 749.0 851.6 858.0 958.1
391.9 437.3 507.1 505.8 713.3 744.1 859.3 854.9 959.1
391.1 436.9 505.6 198.9 701.0 750.1 866.0 868.2 966.7
391.7 438.5 506.8 505.0 710.7 747.7 859.0 860.4 961.3
0.16% 0.65% -0.24% -1.20% -1.36% -0.49% -0.86% 0.91% 0.56%
Figure 2. log-log plot between the contact radius a and the reduced radius of a silica particle R. The unit of the contact radius a and the reduced radius of a silica particle R is the nanometer. The size of silica particles ranges from 400 to 900 nm. The contact radii are corrected according to Rimai’s description.12
force is large enough to cause the particles to generate a plastic deformation.12-15 A log-log plot between the contact radius a and the reduced radius of a silica particle R is illustrated in Figure 2. The unit of the contact radius a and the reduced radius of a silica particle R is the nanometer. The size of silica particles ranges from 400 to 900 nm. The contact radii were corrected for the effect of the Pt coating according to Rimai’s description.12 The least-squares-fit line (Figure 2) reveals that the power-law dependence between the contact radius a and the reduced radius of silica particles R is 0.46 ( 0.08 with a 95% confidence internal. It is not consistent with the JKR or DMT theory, which assumed elastic deformation and predicted the 2/3 power dependence between the particles. However, it is in reasonable agreement with the MP model, which assumed plastic deformation and predicted the 1/2 power dependence. Thus, silica particles may undergo plastic deformation as a result of the surface force. According to the arguments of Maugis and Pollock for plastic deformation, the contact radius under zero load was given in the MP model7 as a ¼
2wa 3Y
The unit of the contact radius a and the reduced radius of silica particles R is the nanometer. The size of silica particles ranges from 400 to 900 nm. The contact radii are corrected according to Rimai’s description.12
be calculated using eq 1 from the MP model. As shown in Figure 3, the least-squares-fit line has a slope of 8.18. The work of adhesion calculated from the slope of the curve is 1.10 J/m2. This is a reasonable value for this system. Moreover, the least-squaresfit line through the points also intercepts the origin within experimental error. This implies the absence of any externally applied load. Similarly, assuming the occurrence of elastic deformation, the work of adhesion can also be calculated by the models of elastic deformation. First, we determine which model of elastic deformations is appropriate to describe our system by the MYD model.6 This model introduced a dimensionless parameter μ to distinguish the ranges of each model. The JKR model is generally applicable to the μ . 1 systems, and the DMT model is valid for the μ , 1 systems. Dimensionless parameter μ is given by " #1=3 32 2Rðwa Þ2 μ ¼ 3π πKz0 3
R1=2
ð1Þ
wa ¼ γ1 þ γ2 -γ12
ð2Þ
where γ1 and γ2 are the surface energies of the two particles and γ12 is their interfacial energy. We testify to the correctness of the MP model for our experimental data from the aspect of the work of adhesion. Assuming the occurrence of plastic deformation, the work of adhesion can
ð3Þ
where K is given by
1=2
where a is the contact radius, R is the reduced radius of particles, Y is the yield point of the material, and wa is the work of adhesion, which can be obtained7 from
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Figure 3. Linear relationship between contact radius a and R0.5.
K ¼ "
4
1 -v1 1 -v2 2 þ 3 E1 E2 2
# -1
ð4Þ
where E1, E2, v1, and v2 are the Young’s moduli and Poisson’s radii of two contact materials, respectively. By substituting E1 = E2 = 72.9 GPa18 and v1 = v2 = 0.17 18 into eq 4 and wa = 1.10 J/m2, R = 400 nm, and z0 = 0.4 nm into eq 3, we calculate dimensionless parameter μ to be 17 123. Thus, the JKR model is appropriate to describe our system. Similarly, the contact radius is plotted as a (18) Gray, D. E. American Institute of Physics Handbook; McGraw-Hill: New York, 1972.
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function of the particle radius to the 2/3 power in Figure 4. As shown in Figure 4, the least-squares-fit line has a slope of 2.67. The work of adhesion values calculated from the slope of the curve using eq 5 (JKR model) and eq 6 (DMT model) are 5.04 and 15.11 J/m2, respectively. The surface energy of the silica particle is 1.0 J/m2.19 Because the work of adhesion is simply the sum of the surface energies of the two contacting materials minus the interfacial energy (as shown in eq 2) and their interfacial energy is zero, the work of adhesion of two contacting silica spheres is 2 J/m2. Thus, 5.04 J/m2 is unrealistically large; 15.11 J/m2 is even more unrealistic. Thus, the JKR and DMT models are not applicable to this system. The value of 1.10 J/m2 (calculated by the MP model) compared to 2 J/m2 is quite small, which may result from experimental errors. The reasons are still under further investigation. The work of adhesion calculated by the MP, JKR, and DMT models is listed in Table 2. These results lend further strength to the argument that plastic deformation may occur. a3 ¼
6πwa R2 K
ð5Þ
a3 ¼
2πwa R2 K
ð6Þ
We discuss the possibility of the occurrence of the plastic deformation for silica particles from the point of view of mechanics. The pressure exerted by the particle on another one could be determined20 by p ¼ 2wa =z0
ð7Þ
where wa is the work of adhesion and z0 is the separation distance between the particles; a value of 0.4 nm is often used. Thus, the pressure exerted on the contact area is approximately 5 109 N/m2. It is much larger than the yield point of the silica (19) Bass, M. Handbook of Optics; McGraw-Hill: New York, 1995. (20) Rimai, D. S.; Moore, R. S.; Bowen, R. C.; Smith, V. K.; Woodgate, P. E. J. Mater. Res. 1993, 8, 662. (21) Wu, X.; Sacher, E.; Meunier, M. J. Appl. Phys. 1999, 86, 1744.
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Figure 4. Linear plot of the contact radius a as a function of the particle to the 2/3 power (R2/3).
Table 2. Work of Adhesion Calculated by the MP, JKR, and DMT Models model wa (J/m2)
MP
JKR
DMT
1.10
5.04
15.11
(1.1 10 N/m ). Therefore, plastic deformation did occur for the contacted silica spheres in the size range of 400-900 nm. Traditionally, the harder the particle, the smaller both the contact radius and the adhesion force. Therefore, it is widely accepted that silica particles should undergo elastic deformation,21 but our experiments give the opposite results. 7
2 19
Conclusions A silica sphere in the size range of 400-900 nm was in contact with that of a similar size. It was found that the silica sphere deformed because of the surface force. The contact radius varies as the reduced radius to the 0.46 ( 0.08 power, which shows reasonable agreement with MP theory. This suggests that silica spheres, as rigid particles, underwent plastic deformation rather than elastic deformation. Acknowledgment. This work was supported by the National Natural Science Foundation of China (grant no. 10825521), by the National 863 Program, and by the Shanghai Committee of Science and Technology, China (grant no. 07DZ22302).
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