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According to the existing shear modulus-pair potential relationship model for colloidal crystal comprised of highly charged colloidal particles, the calculated shear moduli of colloidal crystals are much larger than the measured values by the torsional resonance spectroscopy (TRS). Moreover, by using the relationship model, the effective surface charge of colloidal particles, obtained by fitting values of shear moduli measured by TRS (effective elasticity charge), is smaller than that obtained through the experimental method of conductivity-number density relationship (effectively transported charge). So far there has been no practical explanation to this discrepancy. Our analysis shows that this discrepancy is because the existing relationship model is for the perfect crystals and does not include the defects such as voids which can result in the decrease of mechanical properties of materials. The existing shear modulus-pair potential model will be improved by introducing the effect of voids, which is inspired from the Gibson-Ashby model in the study of cellular solid. The Yukawa potential, which considers Coulomb repulsions between colloidal particles and is usually used in the model expressions, will be substituted by Sogami-Ise potential, which considers a long-range attraction in addition to that Coulomb repulsions and accepts the existence of voids inside the colloidal crystals. For five different kinds of highly charged colloidal particles, the shear moduli with different volume fractions are measured by TRS. Then the fitted effective surface charges using the original and improved model respectively are compared with each other. It can be concluded that the effective elastic charge obtained by the improved model is more suitable and much closer to the renormalized charge obtained from Alexander's method. It is also clear that neither the effectively transported charge nor the Alexander's renormalized charge can be used to evaluate the shear moduli of colloidal crystals with voids inside. These results can also let us further understand and use the effective surface charge in the colloid studies.
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Keywords:
- effective charge /
- shear modulus /
- voids /
- colloidal crystal
[1] Denton A R 2010 J. Phys.-Condens. Matter 22 364108
[2] Zhou H W, Mi L, Liu L X, Xu S H, Sun Z W 2013 Acta Phys. Sin. 62 134704 (in Chinese) [周宏伟, 米利, 刘丽霞, 徐升华, 孙祉伟 2013 物理学报 62 134704]
[3] Alexander S, Chaikin P M, Grant P, Morales G J, Pincus P, Hone D 1984 J. Chem. Phys. 80 5776
[4] Grosse C, Shilov V N 2000 J. Colloid Interf. Sci. 225 340
[5] Palberg T, Schweinfurth H, Kller T, Mller H, Schpe H J, Reinmller A 2013 Eur. Phys. J.-Spec. Top. 222 2835
[6] Ito K, Sumaru K, Ise N 1992 Phys. Rev. B 46 3105
[7] Ouyang W, Zhou H, Xu S, Sun Z 2014 Colloid. Surface A 441 598
[8] Zhou H, Xu S, Ouyang W, Sun Z, Liu L 2013 J. Chem. Phys. 139 064904
[9] Gong Y K, Nakashima K, Xu R 2001 Langmuir 17 2889
[10] Belloni L 1998 Colloid. Surface A 140 227
[11] Wette P, Schpe H J, Palberg T 2002 J. Chem. Phys. 116 10981
[12] Shapran L, Medebach M, Wette P, Palberg T, Schpe H J, Horbach J, Kreer T, Chatterji A 2005 Colloid. Surface A 270-271 220
[13] Wette P, Schpe H J, Palberg T 2003 Colloid. Surface A 222 311
[14] Dubois-Violette E, Pieranski P, Rothen F, Strzelecki L 1980 J. Phys. France 41 369
[15] Joanny J F 1979 J. Colloid Interf. Sci. 71 622
[16] Yoshida H, Ito K, Ise N 1991 Phys. Rev. B 44 435
[17] Zhou H, Xu S, Sun Z, Du X, Liu L 2011 Langmuir 27 7439
[18] Zhou H, Xu S, Sun Z, Zhu R 2015 J. Chem. Phys. 143 144903
[19] Hashin Z, Shtrikman 1962 J. Mech. Phys. Solids 10 343
[20] Zeller R, Dederichs P 1973 Phys. Status Solidi B 55 831
[21] Anderson V J, Terentjev E M, Meeker S P 2001 Eur. Phys. J. E 4 11
[22] Anderson V J, Terentjev E M 2001 Eur. Phys. J. E 4 21
[23] Ashby M F, Medalist R F M 1983 Metall. Trans. A 14 1755
[24] Nieh T, Kinney J, Wadsworth J, Ladd A 1998 Scripta Mater. 38 1487
[25] Ise N, Konishi T, Tata B V R 1999 Langmuir 15 4176
[26] Stevens M J, Falk M L, Robbins M O 1996 J. Chem. Phys. 104 5209
[27] Tata B V R, Ise N 1996 Phys. Rev. B 54 6050
[28] Sogami I, Ise N 1984 J. Chem. Phys. 81 6320
[29] Wang Q, Fu S, Yu T 1994 Prog. Polym. Sci. 19 703
[30] Du X, Xu S H, Sun Z W, Liu L 2012 Chin. J. Chem. Phys. 25 318
[31] Shouldice G T D, Vandezande G A, Rudin A 1994 Eur. Polym. J. 30 179
[32] Goldburg W I 1999 Am. J. Phys. 67 1152
[33] Xiong B, Pallandre A, le Potier I, Audebert P, Fattal E, Tsapis N, Barratt G, Taverna M 2012 Anal. Methods 4 183
[34] Qin Y M, Zhou H W, Xu S H, Sun Z W 2015 Chem. J. Chinese Univ. 36 310 (in Chinese) [秦艳铭, 周宏伟, 徐升华, 孙祉伟 2015 高等学校化学学报 36 310]
[35] Trizac E, Bocquet L, Aubouy M, von Grnberg H H 2003 Langmuir 19 4027
[36] Hessinger D, Evers M, Palberg T 2000 Phys. Rev. E 61 5493
[37] Joanicot M, Jorand M, Pieranski P, Rothen F 1984 J. Phys. France 45 1413
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[1] Denton A R 2010 J. Phys.-Condens. Matter 22 364108
[2] Zhou H W, Mi L, Liu L X, Xu S H, Sun Z W 2013 Acta Phys. Sin. 62 134704 (in Chinese) [周宏伟, 米利, 刘丽霞, 徐升华, 孙祉伟 2013 物理学报 62 134704]
[3] Alexander S, Chaikin P M, Grant P, Morales G J, Pincus P, Hone D 1984 J. Chem. Phys. 80 5776
[4] Grosse C, Shilov V N 2000 J. Colloid Interf. Sci. 225 340
[5] Palberg T, Schweinfurth H, Kller T, Mller H, Schpe H J, Reinmller A 2013 Eur. Phys. J.-Spec. Top. 222 2835
[6] Ito K, Sumaru K, Ise N 1992 Phys. Rev. B 46 3105
[7] Ouyang W, Zhou H, Xu S, Sun Z 2014 Colloid. Surface A 441 598
[8] Zhou H, Xu S, Ouyang W, Sun Z, Liu L 2013 J. Chem. Phys. 139 064904
[9] Gong Y K, Nakashima K, Xu R 2001 Langmuir 17 2889
[10] Belloni L 1998 Colloid. Surface A 140 227
[11] Wette P, Schpe H J, Palberg T 2002 J. Chem. Phys. 116 10981
[12] Shapran L, Medebach M, Wette P, Palberg T, Schpe H J, Horbach J, Kreer T, Chatterji A 2005 Colloid. Surface A 270-271 220
[13] Wette P, Schpe H J, Palberg T 2003 Colloid. Surface A 222 311
[14] Dubois-Violette E, Pieranski P, Rothen F, Strzelecki L 1980 J. Phys. France 41 369
[15] Joanny J F 1979 J. Colloid Interf. Sci. 71 622
[16] Yoshida H, Ito K, Ise N 1991 Phys. Rev. B 44 435
[17] Zhou H, Xu S, Sun Z, Du X, Liu L 2011 Langmuir 27 7439
[18] Zhou H, Xu S, Sun Z, Zhu R 2015 J. Chem. Phys. 143 144903
[19] Hashin Z, Shtrikman 1962 J. Mech. Phys. Solids 10 343
[20] Zeller R, Dederichs P 1973 Phys. Status Solidi B 55 831
[21] Anderson V J, Terentjev E M, Meeker S P 2001 Eur. Phys. J. E 4 11
[22] Anderson V J, Terentjev E M 2001 Eur. Phys. J. E 4 21
[23] Ashby M F, Medalist R F M 1983 Metall. Trans. A 14 1755
[24] Nieh T, Kinney J, Wadsworth J, Ladd A 1998 Scripta Mater. 38 1487
[25] Ise N, Konishi T, Tata B V R 1999 Langmuir 15 4176
[26] Stevens M J, Falk M L, Robbins M O 1996 J. Chem. Phys. 104 5209
[27] Tata B V R, Ise N 1996 Phys. Rev. B 54 6050
[28] Sogami I, Ise N 1984 J. Chem. Phys. 81 6320
[29] Wang Q, Fu S, Yu T 1994 Prog. Polym. Sci. 19 703
[30] Du X, Xu S H, Sun Z W, Liu L 2012 Chin. J. Chem. Phys. 25 318
[31] Shouldice G T D, Vandezande G A, Rudin A 1994 Eur. Polym. J. 30 179
[32] Goldburg W I 1999 Am. J. Phys. 67 1152
[33] Xiong B, Pallandre A, le Potier I, Audebert P, Fattal E, Tsapis N, Barratt G, Taverna M 2012 Anal. Methods 4 183
[34] Qin Y M, Zhou H W, Xu S H, Sun Z W 2015 Chem. J. Chinese Univ. 36 310 (in Chinese) [秦艳铭, 周宏伟, 徐升华, 孙祉伟 2015 高等学校化学学报 36 310]
[35] Trizac E, Bocquet L, Aubouy M, von Grnberg H H 2003 Langmuir 19 4027
[36] Hessinger D, Evers M, Palberg T 2000 Phys. Rev. E 61 5493
[37] Joanicot M, Jorand M, Pieranski P, Rothen F 1984 J. Phys. France 45 1413
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