PHYSICAL REVIEW E 91, 032307 (2015)
Shear-stress function approach of hydration layer based on the Green-Kubo formula Bongsu Kim, Soyoung Kwon, Geol Moon, and Wonho Jhe Centerfor THz-Bio Application Systems and Institute of Applied Physics, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea (Received 5 December 2014; published 16 March 2015) We present the analytic expression of the stress correlation (SC) function for the ubiquitous hydration water layer (HWL) using the Green-Kubo equation and the shear modulus of HWL. The SC function is then experimentally obtained by measuring the viscoelastic properties of HWL using shear-mode dynamic force spectroscopy. Interestingly, the SC changes sign from positive to negative as the HWL thickness increases, where the shear stresses acting on the HWLs bound to two nearby surfaces are out of phase. We also suggest that the repulsive hydration force originates from the SC of HWL. Our results provide the first demonstration of the microscopic understanding of the HWL viscoelasticity and may allow a deeper insight on the HWL dynamics as well as the complex liquids. DOI: 10.1103/PhysRevE.91.032307
PACS number(s): 83.85.Vb, 68.08.-p, 62.10,+s, 05.70.Ln
I. INTRODUCTION
In general, most of fluids exhibit nonequilibrium (NE) characteristics and there have been numerous researches to address their quantitative dynamics for over a century [1,2]. However, it is still challenging to have a full understanding of liquids and condensed gases in the NE states [1] in contrast to the dilute gas [2]. The NE liquids have been usually investigated by the linear response theory based on the GreenKubo equation that provides the relation between the transport coefficients and the correlation function of fluctuations [1,3], whereas the dilute gas has been studied by the kinetic theory based on the Boltzmann equation [2,4], For the complex liquids, where the intermolecular interactions play a critical role, the Green-Kubo equation predicts the shear viscosity (r/) given by tl
= k b
f
d t (a (0)o (t)).
(1)
T Jo
Here V, k B, T, t, and a are the system volume, Boltzmann constant, temperature, time, and shear stress, respectively. The bracket denotes the equilibrium ensemble average. Notice that V / k BT in Eq. (1) can be replaced with \ / k BT V when a represents the stress tensor of a total system volume [5]. Com monly, viscosity becomes a property independent of the liquid volume and V in Eq. (1) can be eliminated by the ensemble average term in the numerator of Eq. (1). Equation (1) indicates that the macroscopically measurable shear viscosity can be calculated by microscopic quantities. The well-developed method to solve Eq. (1) is the molecular dynamic (MD) simulation that evaluates the correlation function with respect to particle velocity and interparticle potential energy [5], Various rheological properties have been studied by MD simulation for simple liquids that can be described by isotropic pair potentials [5-8]. However, in the case of complex liquids, substantially increased simulation lime is needed to compute the quantities such as the complex interparticle potentials and inner molecular motions [9]. Therefore, if the relaxation time of correlation function is longer than the total calculation
’[email protected] 1539-3755/2015/91 (3)/032307(6)
time, it is difficult to predict the rheological properties by MD simulation. The hydration water layer (HWL) is a typical example of complex fluids that possess a long relaxation time [ 10-
12],
The HWL is a ubiquitous form of nanoscale water that consists of water molecules bound to the hydrophilic surfaces [12], nanoparticles [13], and ions [14], It is well known that the HWL plays an important role in diverse phenomena in nature, including self-assembly [13,15], friction [16], and biological processes [17,18], and it has been studied by various methods. Macroscopically, experiments based on surface force apparatus [10,16] and scanning probe microscope [11,12] have reported that the rheological properties of HWL, such as the viscosity and the relaxation time of shear motion, differ from those of bulk water. Microscopically, the MD simulations have revealed that the relevant dynamics, such as the hydrogen-bond lifetime and the dipole autocorrelation, becomes sluggish [18-20], For example, the macroscopic optical absorption coefficients can be calculated by the linear response theory when applied to the microscopic dipole autocorrelation [18]. In this case, the relaxation time of dipole autocorrelation is sufficiently short (10 ps ~ 1 ns) compared to the presently available calculation time of MD simulation [19,20], and therefore the terahertz absorption experiments by HWL can be analyzed in terms of the microscopic water molecular behaviors. However, since the measured relaxation time of HWL with respect to shear deformation is between 1 pts and 10 ms [10-12], it is difficult to use MD simulation for shear dynamics study of HWL, and thus alternatively, we seek the analytic form of Eq. (1) for better understanding of the shear properties of HWL. In this article, we first derive the analytic SC function of HWL by using both the Green-Kubo equation [1,3] with the linear Maxwell model [21] and the shear modulus obtained from the general HWL stress tensor [ 12]. We construct the SC function of HWL whose spatial correlation is included so as to incorporate the fact that the viscosity (or shear modulus) of HWL depends on its thickness. We then show the SC function of HWL can be obtained with the experiment that measures the viscoelastic quantities of HWL by dynamic shear-mode operation of amplitude-modulation atomic force microscopy (AM-AFM). In particular, we find that the spatial part of the
032307-1
©2015 American Physical Society
BONGSU KIM, SOYOUNG KWON, GEOL MOON, AND WONHO JHE
SC function can have negative values when the thickness of HWL is larger than the decay length of the hydration force, while the spatially independent part simply decays as e~'lx (r is the relaxation time). Notice that the negative SC means that the shear stresses on the HWLs of two nearby hydrophilic surfaces act in the opposite direction, which results from the weakly bound HWL that exists between the two strongly bound HWLs [18,22], Moreover, we suggest that the SC of HWL is closely related to the physical origin of the exponentially decaying hydration force because the SC function exhibits the empirically known exponential hydration-force form [12,14].
PHYSICAL REVIEW E 91, 032307 (2015)
where G is the shear modulus. Equation (4) indicates that to calculate the SC function of HWL, we need to know the shear modulus G of HWL. For this, we use the established results of the general stress tensor for HWL, which is based on the empirically known averaged form of the exponentially decaying hydration force. In this formalism, HWL does not oscillate spatially (k = 0), and horizontally homogeneous but vertically inhomogeneous. Moreover, since the linear Maxwell model is applicable to HWL due to its viscoelasticity [10-12], Eq. (4) is valid for HWL where G becomes a function of the thickness of HWL. Therefore, from Eq. (4), G can be expressed in terms of G' or G" as follows:
II. THEORETICAL MODEL
G,j _ + « V _ G„ 1 + co2z 2
The SC function of viscoelastic liquids whose viscosity (or shear modulus G) is independent of their shape or geometry can be obtained by the linear Maxwell model including the following random fluctuation, (o-(O)a(f)) =
(2)
where r is the relaxation time for the SC function [23]. On the other hand, since the viscosity of HWL discussed in this paper depends on its thickness, Eq. (2) cannot be directly applied to the HWL. Equation (1) is applicable when the departure from the equilibrium state is spatially homogeneous and temporally stationary. Therefore, when such a deviation is small enough for the linear analysis to be valid and the liquids system involves spatially or temporally small periodic changes Eq. (1) can be generalize as [24,25] n*
= ~k & i Jfv dr Jo[
d t ( a m M r , t ) ) e iAFMh due to the dominant capillary effects for z > 2.5 nm, whose exponential decay length is 8 nm in Fig. 2. These increased £ afm and /?afm result in the fact that the measured Cn does not quickly converge to zero. The black crosses converge faster than the red empty circles because the silica tip has a smaller interacting area, which results in the weak capillary effects. The blue dashed line in Fig. 4 corresponds to Cn that includes intuitively the additional contribution of the capillary effects, given by the usual exponential form of Cq( 1 —z/X\)e~z^X2, where Co, Xj, and X2 are 0.25, 2.5 nm, and 8 nm, respectively. Another intriguing result is that the SC function, Eq. (12), includes the experimentally well-known hydration force form, Fh = f2P(,e~z^x° [12,14]. This suggests that the SC of HWL and the hydration force may be closely related with the common physical mechanism. For decades, this exponential hydration force has been frequently observed in various experimental situations [10,14,16,30], but nevertheless the origin of the force is still under debate [14,31]. Interestingly, we observe that Eq. (12) expresses the similar behaviors between the hydration force and the SC function, which indicates that the microscopic SC of HWL can be associated with the origin of the empirical hydration force. V. CONCLUSION
In conclusion, we have presented the analytic expression of the SC function of HWL using the Green-Kubo equation with the linear Maxwell model and the shear modulus of HWL derived from the stress tensor of HWL and showed that the viscous and elastic quantities of HWL measured by the shear-mode AM-AFM can be converted to the experimental SC function of HWL. In particular, it is interesting to find that the SC function becomes negative above 1.4 nm of z, which indicates that the shear stresses between two closely neighboring HWLs can act on average in the opposite direc tion. We also suggest that SC can be attributed to produce the hydration force because its functional form behaves similarly to the hydration force form of HWL. Our results provide the microscopic understanding of the viscoelastic HWL or the general complex fluids in terms of the SC function, and, consequently, it may contribute to a deeper statistical study of the nonequilibrium liquids. ACKNOWLEDGMENTS
This work was supported by the National Research Foun dation of Korea (NRF) grant funded by the Korea government (MSIP) (Grant No. 2009-0083512).
[2] L. D. Landau and E. M. Lifshitz, Physical Kinetics (Pergamon, New York, 1981). [3] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
032307-5
BONGSU KIM, SOYOUNG KWON, GEOL MOON, AND WONHO JHE [4] K. Kumar, Aust, J. Phys. 20, 205 (1967). [5] J.-P. Jansen and I. R. McDonald, Theory o f Simple Liquids (Academic Press, London, 2006). [6] J. J. Erpenbeck, Phys. Rev. A. 38, 6255 (1988). [7] G. A. Fernandez, J. Vrabec, and H. Hasse, Fluid Phase Equilibr. 221,157(2004). [8] Z. Donko, J. Goree, and P. Hartmann, Phys. Rev. E 81, 056404 ( 2010).
[9] R. J. Sadus, Molecular Simulation o f Fluids: Theory, Algorithms and Object Orientation (Elsevier, Amsterdam, 1999). [10] Y. Zhu and S. Granick, Phys. Rev. Lett. 87, 096104 (2001). [11] T.-D. Li and E. Riedo, Phys. Rev. Lett. 100, 106102 (2008). [12] B. Kim, Q.H. Kim, S. Kwon, S. An, K. Lee, M. Lee, and W. Jhe, Phys. Rev. Lett. Ill, 246102 (2013). [13] W. Lv and R. Wu, Nanoscale 5, 2765 (2013). [14] J. Israelachvili, Intermolecular and Surface Forces (Academic Press, New York, 2011). [15] T. G. Mazen Ahmad, Wei Gu, and V. Helms, Nat. Commun. 2, 261 (2011). [16] U. Raviv and J. Klein, Science 297, 1540 (2002). [17] J. Israelachvili and H. Wennerstrom, Nature 379, 219 (1996). [18] S. Ebbinghaus, S. J. Kim, M. Heyden, X. Yu, U. Heugen, M. Gruebele, D. M. Leitner, and M. Havenith, Proc. Natl. Acad. Sci. USA 104, 20749 (2013).
PHYSICAL REVIEW E 91, 032307 (2015)
[19] Y. Leng and P. T. Cummings, Phys. Rev. Lett. 94, 026101 (2005). [20] S. R.-V. Castrillon, N. Giovambattista, I. A. Aksay, and P. G. Debenedetti, J. Phys. Chem. B 113, 7973 (2009). [21] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics o f Polymeric Liquids: Volume 1 Fluid Mechanics (WileyInterscience, New York, 1987). [22] B. Kim, S. Kwon, H. Mun, S. An, and W. Jhe, Sci. Rep. 4, 6499 (2014). [23] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.91.032307 for mathematical details. [24] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Addison-Wesley, New York, 1983), Chap. 4.6. [25] K. Kawasaki and A. Onuki, Phys. Rev. A 42, 3664 (1990). [26] M. Lee, J. Jahng, K. Kim, and W. Jhe, Appl. Phys. Lett. 91, 023117 (2007). [27] M. Lee, B. Sung, N. Hashemi, and W. Jhe, Faraday Discuss. 141,415(2009). [28] M. Lee and W. Jhe, Phys. Rev. Lett. 97, 036104 (2006). [29] S. Kwon, C. Stambaugh, B. Kim, S. Ana, and W. Jhe, Nanoscale 6, 5474 (2014). [30] G. Peschel, P. Belouschek, M. M. Muller, M. R. Muller, and R. Konig, Colloid Polym. Sci. 260, 444 (1982). [31] J. Faraudo and F. Bresme, Phys. Rev. Lett. 94, 077802 (2005).
032307-6
Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
Shear-stress function approach of hydration layer based on the Green-Kubo formula Bongsu Kim, Soyoung Kwon, Geol Moon, and Wonho Jhe Centerfor THz-Bio Application Systems and Institute of Applied Physics, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea (Received 5 December 2014; published 16 March 2015) We present the analytic expression of the stress correlation (SC) function for the ubiquitous hydration water layer (HWL) using the Green-Kubo equation and the shear modulus of HWL. The SC function is then experimentally obtained by measuring the viscoelastic properties of HWL using shear-mode dynamic force spectroscopy. Interestingly, the SC changes sign from positive to negative as the HWL thickness increases, where the shear stresses acting on the HWLs bound to two nearby surfaces are out of phase. We also suggest that the repulsive hydration force originates from the SC of HWL. Our results provide the first demonstration of the microscopic understanding of the HWL viscoelasticity and may allow a deeper insight on the HWL dynamics as well as the complex liquids. DOI: 10.1103/PhysRevE.91.032307
PACS number(s): 83.85.Vb, 68.08.-p, 62.10,+s, 05.70.Ln
I. INTRODUCTION
In general, most of fluids exhibit nonequilibrium (NE) characteristics and there have been numerous researches to address their quantitative dynamics for over a century [1,2]. However, it is still challenging to have a full understanding of liquids and condensed gases in the NE states [1] in contrast to the dilute gas [2]. The NE liquids have been usually investigated by the linear response theory based on the GreenKubo equation that provides the relation between the transport coefficients and the correlation function of fluctuations [1,3], whereas the dilute gas has been studied by the kinetic theory based on the Boltzmann equation [2,4], For the complex liquids, where the intermolecular interactions play a critical role, the Green-Kubo equation predicts the shear viscosity (r/) given by tl
= k b
f
d t (a (0)o (t)).
(1)
T Jo
Here V, k B, T, t, and a are the system volume, Boltzmann constant, temperature, time, and shear stress, respectively. The bracket denotes the equilibrium ensemble average. Notice that V / k BT in Eq. (1) can be replaced with \ / k BT V when a represents the stress tensor of a total system volume [5]. Com monly, viscosity becomes a property independent of the liquid volume and V in Eq. (1) can be eliminated by the ensemble average term in the numerator of Eq. (1). Equation (1) indicates that the macroscopically measurable shear viscosity can be calculated by microscopic quantities. The well-developed method to solve Eq. (1) is the molecular dynamic (MD) simulation that evaluates the correlation function with respect to particle velocity and interparticle potential energy [5], Various rheological properties have been studied by MD simulation for simple liquids that can be described by isotropic pair potentials [5-8]. However, in the case of complex liquids, substantially increased simulation lime is needed to compute the quantities such as the complex interparticle potentials and inner molecular motions [9]. Therefore, if the relaxation time of correlation function is longer than the total calculation
’[email protected] 1539-3755/2015/91 (3)/032307(6)
time, it is difficult to predict the rheological properties by MD simulation. The hydration water layer (HWL) is a typical example of complex fluids that possess a long relaxation time [ 10-
12],
The HWL is a ubiquitous form of nanoscale water that consists of water molecules bound to the hydrophilic surfaces [12], nanoparticles [13], and ions [14], It is well known that the HWL plays an important role in diverse phenomena in nature, including self-assembly [13,15], friction [16], and biological processes [17,18], and it has been studied by various methods. Macroscopically, experiments based on surface force apparatus [10,16] and scanning probe microscope [11,12] have reported that the rheological properties of HWL, such as the viscosity and the relaxation time of shear motion, differ from those of bulk water. Microscopically, the MD simulations have revealed that the relevant dynamics, such as the hydrogen-bond lifetime and the dipole autocorrelation, becomes sluggish [18-20], For example, the macroscopic optical absorption coefficients can be calculated by the linear response theory when applied to the microscopic dipole autocorrelation [18]. In this case, the relaxation time of dipole autocorrelation is sufficiently short (10 ps ~ 1 ns) compared to the presently available calculation time of MD simulation [19,20], and therefore the terahertz absorption experiments by HWL can be analyzed in terms of the microscopic water molecular behaviors. However, since the measured relaxation time of HWL with respect to shear deformation is between 1 pts and 10 ms [10-12], it is difficult to use MD simulation for shear dynamics study of HWL, and thus alternatively, we seek the analytic form of Eq. (1) for better understanding of the shear properties of HWL. In this article, we first derive the analytic SC function of HWL by using both the Green-Kubo equation [1,3] with the linear Maxwell model [21] and the shear modulus obtained from the general HWL stress tensor [ 12]. We construct the SC function of HWL whose spatial correlation is included so as to incorporate the fact that the viscosity (or shear modulus) of HWL depends on its thickness. We then show the SC function of HWL can be obtained with the experiment that measures the viscoelastic quantities of HWL by dynamic shear-mode operation of amplitude-modulation atomic force microscopy (AM-AFM). In particular, we find that the spatial part of the
032307-1
©2015 American Physical Society
BONGSU KIM, SOYOUNG KWON, GEOL MOON, AND WONHO JHE
SC function can have negative values when the thickness of HWL is larger than the decay length of the hydration force, while the spatially independent part simply decays as e~'lx (r is the relaxation time). Notice that the negative SC means that the shear stresses on the HWLs of two nearby hydrophilic surfaces act in the opposite direction, which results from the weakly bound HWL that exists between the two strongly bound HWLs [18,22], Moreover, we suggest that the SC of HWL is closely related to the physical origin of the exponentially decaying hydration force because the SC function exhibits the empirically known exponential hydration-force form [12,14].
PHYSICAL REVIEW E 91, 032307 (2015)
where G is the shear modulus. Equation (4) indicates that to calculate the SC function of HWL, we need to know the shear modulus G of HWL. For this, we use the established results of the general stress tensor for HWL, which is based on the empirically known averaged form of the exponentially decaying hydration force. In this formalism, HWL does not oscillate spatially (k = 0), and horizontally homogeneous but vertically inhomogeneous. Moreover, since the linear Maxwell model is applicable to HWL due to its viscoelasticity [10-12], Eq. (4) is valid for HWL where G becomes a function of the thickness of HWL. Therefore, from Eq. (4), G can be expressed in terms of G' or G" as follows:
II. THEORETICAL MODEL
G,j _ + « V _ G„ 1 + co2z 2
The SC function of viscoelastic liquids whose viscosity (or shear modulus G) is independent of their shape or geometry can be obtained by the linear Maxwell model including the following random fluctuation, (o-(O)a(f)) =
(2)
where r is the relaxation time for the SC function [23]. On the other hand, since the viscosity of HWL discussed in this paper depends on its thickness, Eq. (2) cannot be directly applied to the HWL. Equation (1) is applicable when the departure from the equilibrium state is spatially homogeneous and temporally stationary. Therefore, when such a deviation is small enough for the linear analysis to be valid and the liquids system involves spatially or temporally small periodic changes Eq. (1) can be generalize as [24,25] n*
= ~k & i Jfv dr Jo[
d t ( a m M r , t ) ) e iAFMh due to the dominant capillary effects for z > 2.5 nm, whose exponential decay length is 8 nm in Fig. 2. These increased £ afm and /?afm result in the fact that the measured Cn does not quickly converge to zero. The black crosses converge faster than the red empty circles because the silica tip has a smaller interacting area, which results in the weak capillary effects. The blue dashed line in Fig. 4 corresponds to Cn that includes intuitively the additional contribution of the capillary effects, given by the usual exponential form of Cq( 1 —z/X\)e~z^X2, where Co, Xj, and X2 are 0.25, 2.5 nm, and 8 nm, respectively. Another intriguing result is that the SC function, Eq. (12), includes the experimentally well-known hydration force form, Fh = f2P(,e~z^x° [12,14]. This suggests that the SC of HWL and the hydration force may be closely related with the common physical mechanism. For decades, this exponential hydration force has been frequently observed in various experimental situations [10,14,16,30], but nevertheless the origin of the force is still under debate [14,31]. Interestingly, we observe that Eq. (12) expresses the similar behaviors between the hydration force and the SC function, which indicates that the microscopic SC of HWL can be associated with the origin of the empirical hydration force. V. CONCLUSION
In conclusion, we have presented the analytic expression of the SC function of HWL using the Green-Kubo equation with the linear Maxwell model and the shear modulus of HWL derived from the stress tensor of HWL and showed that the viscous and elastic quantities of HWL measured by the shear-mode AM-AFM can be converted to the experimental SC function of HWL. In particular, it is interesting to find that the SC function becomes negative above 1.4 nm of z, which indicates that the shear stresses between two closely neighboring HWLs can act on average in the opposite direc tion. We also suggest that SC can be attributed to produce the hydration force because its functional form behaves similarly to the hydration force form of HWL. Our results provide the microscopic understanding of the viscoelastic HWL or the general complex fluids in terms of the SC function, and, consequently, it may contribute to a deeper statistical study of the nonequilibrium liquids. ACKNOWLEDGMENTS
This work was supported by the National Research Foun dation of Korea (NRF) grant funded by the Korea government (MSIP) (Grant No. 2009-0083512).
[2] L. D. Landau and E. M. Lifshitz, Physical Kinetics (Pergamon, New York, 1981). [3] R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
032307-5
BONGSU KIM, SOYOUNG KWON, GEOL MOON, AND WONHO JHE [4] K. Kumar, Aust, J. Phys. 20, 205 (1967). [5] J.-P. Jansen and I. R. McDonald, Theory o f Simple Liquids (Academic Press, London, 2006). [6] J. J. Erpenbeck, Phys. Rev. A. 38, 6255 (1988). [7] G. A. Fernandez, J. Vrabec, and H. Hasse, Fluid Phase Equilibr. 221,157(2004). [8] Z. Donko, J. Goree, and P. Hartmann, Phys. Rev. E 81, 056404 ( 2010).
[9] R. J. Sadus, Molecular Simulation o f Fluids: Theory, Algorithms and Object Orientation (Elsevier, Amsterdam, 1999). [10] Y. Zhu and S. Granick, Phys. Rev. Lett. 87, 096104 (2001). [11] T.-D. Li and E. Riedo, Phys. Rev. Lett. 100, 106102 (2008). [12] B. Kim, Q.H. Kim, S. Kwon, S. An, K. Lee, M. Lee, and W. Jhe, Phys. Rev. Lett. Ill, 246102 (2013). [13] W. Lv and R. Wu, Nanoscale 5, 2765 (2013). [14] J. Israelachvili, Intermolecular and Surface Forces (Academic Press, New York, 2011). [15] T. G. Mazen Ahmad, Wei Gu, and V. Helms, Nat. Commun. 2, 261 (2011). [16] U. Raviv and J. Klein, Science 297, 1540 (2002). [17] J. Israelachvili and H. Wennerstrom, Nature 379, 219 (1996). [18] S. Ebbinghaus, S. J. Kim, M. Heyden, X. Yu, U. Heugen, M. Gruebele, D. M. Leitner, and M. Havenith, Proc. Natl. Acad. Sci. USA 104, 20749 (2013).
PHYSICAL REVIEW E 91, 032307 (2015)
[19] Y. Leng and P. T. Cummings, Phys. Rev. Lett. 94, 026101 (2005). [20] S. R.-V. Castrillon, N. Giovambattista, I. A. Aksay, and P. G. Debenedetti, J. Phys. Chem. B 113, 7973 (2009). [21] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics o f Polymeric Liquids: Volume 1 Fluid Mechanics (WileyInterscience, New York, 1987). [22] B. Kim, S. Kwon, H. Mun, S. An, and W. Jhe, Sci. Rep. 4, 6499 (2014). [23] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.91.032307 for mathematical details. [24] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Addison-Wesley, New York, 1983), Chap. 4.6. [25] K. Kawasaki and A. Onuki, Phys. Rev. A 42, 3664 (1990). [26] M. Lee, J. Jahng, K. Kim, and W. Jhe, Appl. Phys. Lett. 91, 023117 (2007). [27] M. Lee, B. Sung, N. Hashemi, and W. Jhe, Faraday Discuss. 141,415(2009). [28] M. Lee and W. Jhe, Phys. Rev. Lett. 97, 036104 (2006). [29] S. Kwon, C. Stambaugh, B. Kim, S. Ana, and W. Jhe, Nanoscale 6, 5474 (2014). [30] G. Peschel, P. Belouschek, M. M. Muller, M. R. Muller, and R. Konig, Colloid Polym. Sci. 260, 444 (1982). [31] J. Faraudo and F. Bresme, Phys. Rev. Lett. 94, 077802 (2005).
032307-6
Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
