Effect of finite ion sizes in electric double layer mediated interaction force between two soft charged plates

Abstract

We present a theory to compute the electric-double-layer (EDL) mediated per unit area interaction force, or the osmotic pressure, between two soft charged plates separated by a thin layer of electrolyte solution with and without the consideration of finite ion sizes or finite steric effect. These soft charged plates are represented by a charged polyelectrolyte layer (PEL) sandwiched between a rigid plate and an electrolyte solution. The thickness of this PEL is considered to be independent of the EDL effects, rather being governed by the balance of the elastic and the volume exclusion effects of the polyelectrolyte chain. The PEL-rigid-solid interface is considered as uncharged. We provide closed-form analytical results to demonstrate that for a given concentration of electrolyte ions, the osmotic pressure depends solely on two parameters: first, the ratio of the Debye length to a thickness that quantifies the charge content of the PEL, and second, the steric factor quantifying the ion size or steric effect. More importantly, we discover a hidden relationship that connects this ratio of the two thicknesses to the dimensionless Debye length and the dimensionless PEL thickness. This relationship ensures that the osmotic pressure does depend on the plate separation. Finally, we show that the steric effect substantially augments the osmotic pressure, attributed to the corresponding increase in the electrostatic potential at the PEL-rigid-plate interface.

---

1. Introduction

A charged surface, in contact with an electrolyte solution, develops a double layer of charges that screens the bare charge on the surface. This double layer, well known as the Electric Double Layer (EDL),^1,2 dictates a large number of natural and technological phenomena. Experiments and models have been developed to pinpoint the effect of different physical variables that govern the EDL electrostatic potential distribution and the resulting EDL-mediated force. One such thing is the presence of a grafted polyelectrolyte layer (PEL).^3–6 Such a structure, where a PEL is sandwiched between a rigid plate and an electrolyte solution, has been conveniently used to represent a soft charged interface, relevant in a variety of biological and technological scenarios.^7–23 Ohshima and co-workers as well as others have developed several models to quantify this EDL-mediated interaction force between soft charged surfaces. Different cases investigated include the interaction force in presence of ion solubility,²⁴ charge effects²⁵ and interdigitation;²⁶ the interaction force between soft multi-layer spherical and cylindrical particles;^8,27,28 the interaction force for very small separation distance;^29,30 etc.

Here, we provide a free-energy-based model^31,32 to calculate this EDL-mediated interaction force between two soft charged plates with and without the finite ion size effect. This force per unit area is known as the osmotic pressure. The free-energy-based model allows a straightforward calculation of this osmotic pressure from the expression of the corresponding per unit volume free energy.^31,32 The elegance of using this free energy based technique lies in the fact that it becomes trivially simple to account for any new effect that influences the EDL electrostatics – all that is needed is to ensure that this new effect is appropriately accounted for in the free energy expression. There are two key results of this calculation. First, when we disregard the finite ion size effect, meaning that we work in the dilute solution limit,^33–36 we find that for the case where the PEL-rigid-solid interface is uncharged,^3–5 the dimensionless osmotic pressure, made dimensionless by the bulk pressure that is a function of the bulk electrolyte concentration, depends solely on the ratio (denoted as K_λ) of the Debye length to an equivalent thickness that characterizes the charge of the PEL.³⁴ More importantly, however, we discover a hitherto unknown connection between K_λ and the dimensionless Debye length and the dimensionless PEL thickness (made dimensionless with the plate separation distance). This connection, valid only for the case where the PEL-rigid-solid interface is uncharged, ensures that the osmotic pressure does depend on the plate separation. The second and the most important finding is that the consideration of finite ion size effect substantially augments this EDL interaction force. For this case the dependence of the osmotic pressure on the plate separation comes from the dependence of K_λ on the ion size effect, the dimensionless Debye length, and the dimensionless PEL thickness. We were probably among the first groups to study the effect of finite ion size in the EDL distribution at a soft charged interface;³⁴ here we extend this study to quantify the effect of the finite ion sizes in the EDL-mediated interaction force between two soft charged plates.

It is worthwhile here to emphasize the significance of the present study in the context of the applications involving polyelectrolyte-grafted systems. In the present study, we consider that the polyelectrolyte-rigid-solid interface is uncharged. This typically represents a metal–polyelectrolyte interface; therefore, our analysis is of utmost importance in quantifying a large number of applications involving polyelectrolyte-grafted metal systems, e.g., nanoparticle-mediated stabilization of pickering emulsions^37–39 and foams,^37,40,41 preparing crystals that are immune to cytotoxicity,⁴² fabrication of evaporation-mediated nanocomposites with homogeneous particle distribution,⁴³ magnetic nanoparticle induced water harvesting,⁴⁴ quantifying the interactions between cell membrane and polyelectrolyte-grafted nanoparticles for drug delivery,⁴⁵ etc.

Finally, we would like to mention here that here we restrict ourselves within the mean-field-based domain of capturing finite ion size effects. Therefore, we do not account for non-mean-field-based effects like the ion–ion correlation effect, which typically becomes important for any of the following three situations: (a) concentrated electrolytes, (b) multivalent electrolytes, and (c) solvent-free ionic liquids.⁴⁶ Since in the present case, none of these three situations arise, we can justify our use of mean-field-based tracking of ion size effects.

2. Theory

We consider two soft charged plates separated by a thin layer of electrolyte solution, as shown in Fig. 1. The PEL, characterizing the soft charged interface, harbours a particular type of ion. These ions, which will henceforth be denoted as PEL ions, remain excluded from the bulk electrolyte solution, although the electrolyte ions can be present both inside and outside the PEL. Such a behavior of the PEL ensures that the PEL-electrolyte interface behaves as a semi-permeable membrane with respect to the PEL ions, and the electrostatic potential deep within the PEL is the Donnan potential. In other words, when the PEL thickness is assumed to be much larger than the Debye length, the potential at the PEL-rigid-solid interface is denoted as the Donnan potential. Here, before embarking on the free-energy-based approach to calculate the osmotic pressure between the soft charged plates, we would like to state the two key assumptions of our model. First, it is assumed that the PEL thickness d is governed by the elastic and the volume exclusion effects of the polyelectrolyte molecules. Here, the elastic effects represent an entropic free energy loss, whereas the volume exclusion effects result from the repulsive monomer–monomer interactions.⁴⁷ Of course, this assumption also implies that that the EDL and the electrostatic effects do not contribute to the selection of d (see Section 4A for more details). Our second assumption is that the grafting density of the PEL is low,⁴⁷ which allows consideration of uniform steric factor and uniform relative permittivity both inside and outside the PEL (see Section 4B for details).

Fig. 1 Schematic of the two soft charged plates separated by an electrolyte solution. Here the PEL ions, which are positively charged and localized within the PEL, are shown in green. On the other hand, the electrolytic cations are shown in blue and the electrolytic anions are shown in red.

Following ref. 34, the free energy F of the system can be expressed as:

F = ∫f(ψ,n_±)d³r, (1)

where f is the free energy density, expressed as:In eqn (2), ε₀ is the permittivity of free space and ε_r is the relative permittivity of the medium, which is assumed to be identical for the media both inside and outside the PEL^3,48 (see Section 4B for more details). Also in eqn (2), ψ is the electrostatic potential, n_± are the ion number densities of the cations and the anions of the electrolyte, e is the electronic charge, z is the valence of the electrolyte ions (the electrolyte is assumed to be symmetric), k_BT is the thermal energy, a is the ion size, N and Z are the number density and the valence of the PEL ions and θ is the heaviside function. Here we assume that the ion size a remains uniform both inside and outside the PEL (see Section 4B for more details), and is the same for both the electrolyte cations and anions. For a more general situation, eqn (2) can be easily extended for the case that considers non-uniform ion sizes; the corresponding contribution for the change in mixing entropy, accounting for the different sizes of the cations and anions, will be expressed in a manner similar to that proposed by Liu and Eisenberg.⁴⁹ Also, the number density N of the PEL ions is considered to be constant. In eqn (2), the first term denotes the self-electrostatic energy associated with the PEL and the EDL, the second term expresses the contribution of the electrostatic energy of the electrolyte ions being present within the electrostatic potential field ψ, the third term provides the contribution of the electrostatic energy of the PEL ions, and the final term provides the contribution of the mixing entropy of the electrolyte ions by appropriately accounting for the steric effects. This standard expression for the free energy, without the contribution of the term describing the electrostatic effects of the PEL ions, has been prescribed in several other studies describing the EDL electrostatics in presence of finite ion size effects.^33,50

The equilibrium describing the system can be obtained by minimizing F with respect to ψ, n₊ and n₋. In order to calculate the osmotic pressure, we first consider a total derivative of f(y) expressed as:³¹

and ξ_i can be either ψ, n₊ or n₋. Using eqn (2) in eqn (3), and integrating the resulting equation, we can write:

This constant appearing in the right hand side of eqn (4) is equal to the negative of the total pressure (P), which is the sum of the osmotic pressure (Π_osm) and the bulk pressure (P_bulk).^31,32 Considering P_bulk = 2k_BTn_∞ (where n_∞ is the bulk number density of the electrolyte ions), we get:

Since there is equilibrium, we shall have:³⁴

where ν = n_∞a³ is the steric factor.³⁴ Since the ion size a is assumed to be uniform inside and outside the PEL, ν is also uniform inside and outside the PEL. For a thin nanochannel, i.e., when the charged soft plates are nanometric distance apart, the EDLs may overlap. For such a case, this n_∞ is the number density of the electrolyte ions (identical for cations and anions) in the reservoirs connecting the nanochannel.⁵¹ Using eqn (6) we can simplify eqn (5) as:

Since we consider equilibrium, Π_osm must be constant across the entire soft nanochannel^32,53 and therefore can be computed at any y. In fact, this constancy across the entire nanochannel makes it physically consistent that the contribution of the PEL charge density, which is restricted only within the PEL, does not appear in the osmotic pressure expression. Please note that the term dictating the PEL charge density will never appear in the osmotic pressure expression, irrespective of whether the charge density is constant or pH-dependent.²³ Computing Π_osm at the PEL-solid interface, where ψ = ψ_D and dψ/dy = 0 (please note that typically this condition of zero first derivative holds at the PEL-rigid-solid interface for a metallic solid) we get:

Following Chanda and Das,³⁴ we can express ψ_D in terms of ν as:

where K_λ = λ_PEL/λ. Here is the Debye length and is an equivalent thickness within the PEL quantifying the PEL charge content.³⁴ For ν → 0, it is easy to see that:

It can also be shown that eqn (10) can be re-written as:

Eqn (11) is a well-known interrelationship between the Donnan potential and the parameter K_λ for the case where the PEL-rigid-solid interface is uncharged.³⁴ Using eqn (9) in eqn (8), we finally get:

which is the closed form analytical expression of the dimensionless osmotic pressure in terms of only two parameters, namely K_λ and ν.

3. Results

A. Osmotic pressure without finite ion size effects

We can express the osmotic pressure analytically for the case where the ion size effect is neglected (i.e., ν → 0) as [using eqn (8) and (9)]:

In Fig. 2, we show the variation of the dimensional osmotic pressure with K_λ without considering finite ion size for different values of ionic concentration. We can simplify eqn (13) at the limiting values of very large and very small K_λ (see Appendix), and these limiting values can be easily seen in Fig. 2. The non-dimensional version of the osmotic pressure is shown in the inset of Fig. 2.

Fig. 2 Variation of the dimensional osmotic pressure with K_λ for different values of ionic concentration c_∞ expressed in moles per liter or M (n_∞ = 10³N_Ac_∞, where N_A is the Avogadro number and n_∞ has the units of m⁻³). In the inset of the plot we provide the variation of dimensionless osmotic pressure ([capital Pi, Greek, macron]_osm) with K_λ. Here we take ν → 0.

Eqn (13) describes a highly non-intuitive situation, where we get the osmotic pressure (without finite ion size effect) as independent of the separation distance between the two charged soft plates. This is completely contradictory to standard understanding of the osmotic pressure between two charged soft plates.^25–30 This motivated us to re-check the validity of our calculations, and what we discovered was amazing. We found that there was no error in our theory; rather there is a problem in prescribing a K_λ value that is independent of the other two dimensionless parameters, namely d̄ (= d/h) and [small lambda, Greek, macron] (= λ/h). In other words, specifying d̄ and [small lambda, Greek, macron] should automatically fix a K_λ value. This most remarkable scenario only occurs for the case where the PEL-solid-interface is uncharged, and it has been completely overlooked in all these years of research on electrostatics of charged soft interfaces. Here we describe this issue in as much detail as possible.

For an uncharged PEL-solid-interface, one can obtain a condition relating K_λ and ψ_s (where ψ_s is the electrostatic potential at the PEL-rigid-solid interface) by considering a local charge electroneutrality at the PEL-solid interface. Following Dukhin and co-workers,^18,52 this condition can be expressed as:

where is the dimensionless form of ψ_s. Please note that eqn (14) is used universally in modeling electrostatics of soft charged surfaces and provides perhaps the most well-known relationship in the topic,^5,18,52 namely:where and ψ_E is the electrostatic potential at the PEL-electrolyte interface. We must obtain numerical solution for the potential distribution ψ, which in turn will provide us and hence K_λ [by using eqn (14)].

This numerical solution for ψ is obtained by solving the following set of equations:

in presence of the conditions:

Fig. 3 provides the numerical result for the variation of K_λ as a function of d̄ and [small lambda, Greek, macron].

Fig. 3 Numerical results for the variation of K_λ with dimensionless PEL thickness d̄ for different values of dimensionless EDL thickness [small lambda, Greek, macron].

B. Osmotic pressure with finite ion size effects

In Fig. 4, we plot the variation of the dimensionless osmotic pressure with K_λ for different values of the steric factor ν. This is the central result of this paper. Presence of the steric effect increases the osmotic pressure, and also ensures that even for large K_λ the osmotic pressure is finite. For a given ν, the osmotic pressure is a single-valued function of the Donnan potential (or the potential ψ_s) [see eqn (8)]; increase in ψ_s necessarily increases the osmotic pressure. From a very naive viewpoint it can be thought of as analogous to the well-known case of an enhanced osmotic pressure for larger wall potentials.³² Furthermore, Chanda and Das³⁴ demonstrated that the Donnan potential increases with the steric effect on account of the lowering of the net electrolyte charge content due to finite ion size effect;³⁵ this explains why we find an augmented osmotic pressure at finite ν values. The important asymptotic limits of [capital Pi, Greek, macron]_osm are derived in the Appendix, and are recovered from Fig. 4.

Fig. 4 Variation of the dimensionless osmotic pressure [[capital Pi, Greek, macron]_osm = Π/(2n_∞k_BT] with K_λ for different values of the steric factor ν.

Just like the case of no Steric effect, here too K_λ cannot be specified independently; rather it will be a function of ν, d̄, and [small lambda, Greek, macron]. In order to obtain this dependence, we would need to solve the following equation numerically (see Chanda and Das³⁴ for the derivation of this equation):

in presence of the boundary conditions expressed in eqn (17). The basis of writing eqn (18) in the above manner is the fact that the no charge condition at the wall yields:³⁴Once [small psi, Greek, macron]_s = [small psi, Greek, macron]|_ȳ=−1 has been obtained, we can obtain the necessary dependence of K_λ on ν, d̄, and [small lambda, Greek, macron] by using eqn (19). Here we refrain from providing the detailed numerical solution, since it adds no new value to the analysis.

4. Discussions

A. Choice of the PEL thickness d

Central to our analysis is the choice of the PEL thickness d. Many studies on modelling the equilibrium dimension of the grafted PEL brush^47,53–57 indicate that this thickness d is obtained by minimizing the corresponding free energy of the system, which consists of contributions such as the elastic and the excluded volume (or the second virial contribution) free energies, the free energies associated with the confining of the counterions, and the electrostatic contribution of the charges of the PEL. Studies such as that by Netz and Andelman⁵⁸ point out the different possibilities of d and its dependence on the polymer size (or degree of polymerization N_p), for different physical situations. Such physical situations can be, for example, the different relative values of the Debye length with respect to the PEL thickness, and they may dictate which of the free energy contributions exert the most dominating influence in selecting d. For our case, we assume that the effects associated strictly with the polymer, namely the entropic elastic effects, or the stretching energy of the chains and the volume exclusion effects, or the effects associated with the steric repulsion between the monomers, dominate other influences. Therefore, the equilibrium d is selected on account of the balance of these two effects, and following ref. 59 we get:where b_k is the Kuhn length and v₀ = b_k³(1 − 2χ) is the excluded volume parameter (χ is the Flory Huggins parameter). The majority of the studies that calculate the electrostatics of the PEL-grafted surfaces assume a constant d and calculate the electrostatic potential distribution accordingly. Therefore, it is implicitly assumed that the electrostatic effects do not contribute to the selection of d, or in other words, d is selected by the balance of the non-electrostatic (i.e., the elastic and the volume exclusion) effects. There is extensive literature on the calculation of the PEL equilibrium height d that is based on the balance of both the non-electrostatic and the electrostatic effects;^53–57 however, several of them suffer from issues such as the exclusion of the contribution of the PEL charge or the lack of consideration of the detailed EDL structure while calculating the equilibrium d.

B. Consideration of uniform steric factor and uniform permittivity for both inside and outside the PEL

Our model is based on the consideration of an identical steric factor for both inside and outside the PEL. The fundamental basis for such an assumption is that the volume V₀ over which the PEL is present is much larger than the actual physical volume of all the PEL molecules (V_PEL). This condition (V₀/V_PEL ≫ 1) effectively implies that the presence of the polyelectrolyte molecules hardly affects the available space of the electrolyte ions within the PEL, thereby justifying the use of identical values of the steric factor both inside and outside the PEL. We shall show here that the condition V₀/V_PEL ≫ 1, and hence the choice of uniform steric factor, remains valid as long as the grafting density of the PEL is low. Considering the PEL thickness as d and the total plate area of the plate as A_P, we can write V₀ = dA_P. Furthermore, considering a polyelectrolyte grafting density of σ⁻¹ (where σ denotes the surface area pertaining to a single chain),⁴⁷ polyelectrolyte chain thickness a_pol and a polyelectrolyte contour length L_c, we get, V_PEL = A_Pa_pol²L_c/σ. Consequently, we get:

Using eqn (20) to express d and considering L_c = N_Pb_k, we shall get from eqn (21)

For most of the standard polymers, b_k ≫ a_pol (e.g., for DNA molecules, b_k ∼ 100 nm and a_pol ∼ 1–2 nm). Also, for weakly grafted systems, following Zhulina et al.,⁴⁷ we always have, σ/a_pol². Hence, from eqn (22), we get V₀/V_PEL ≫ 1. Consequently, we can infer that for a weakly grafted system, the use of uniform steric factor both inside and outside the PEL is valid.

The next important issue is the choice of the relative permittivity. Like the steric factor, the relative permittivity is also considered to be uniform both inside and outside the PEL, stemming from the consideration of the weakly grafted polyelectrolyte system. In other words, weak grafting density and the consequent negligible volume effects of the polyelectrolyte molecules (see the above paragraph) would mean that the relative permittivity of the polyelectrolyte molecule, which is substantially less than that of the surrounding water, would have negligible impact in altering the permittivity inside the PEL. Virtually all studies of electrostatics of polyelectrolyte-grafted system, therefore, have invariably considered uniform permittivity both inside and outside the PEL.

C. Choice of the appropriate boundary condition at the PEL-rigid-solid interface

In terms of the nature of the solid and the corresponding electrostatic boundary conditions at the solid-electrolyte interface, a solid has been described as being of three different types: (1) Non-polarizable solid, (2) ideally polarizable solid, and (3) finitely polarizable solid. For the first case, the zeta potential is considered as a native property of the solid, reflecting the magnitude of the immobile charge density on it.⁶⁰ For the second case, the solid is a metal, and the net charge density at the liquid–solid interface is strictly zero.⁶¹ Finally, for the third case, the electric displacement on either side of the solid–liquid interface is identical. Of course, the third case is most appropriate, since most of the solids are finitely polarizable. Despite that, we choose the boundary condition that pertains to that of an ideally polarizable solid. The primary motivation for such a selection is that we wish to present our results in a framework that has been mostly used for calculations of electrostatics and electrostatics-mediated osmotic pressure for polyelectrolyte-grafted systems.^24,29,62 In other words, we want to employ a framework (i.e., the same geometry, same boundary conditions, etc.) that is consistent with other studies on soft electrostatic calculations.^3–10

5. Conclusions

To summarize, we have developed a free energy based model to compute the osmotic pressure, quantifying the EDL-mediated interaction force, between two soft charged plates in presence and in absence of finite ion size effects. In addition to quantifying the relevant dimensionless parameters that dictate the problem, the central result of the study points to substantial augmentation of the osmotic pressure in presence of finite ion size effects caused by the corresponding enhancement of the Donnan potential. The present calculations provide useful relevant hints at the possible large electrostatic potential mediated EDL interactions that are in action in a myriad of applications in vast number of disciplines, such as drug delivery, enhanced oil recovery, foam stabilization, water harvesting, and many more.

Appendix

A.1. Derivation of the limiting values of osmotic pressure for very large and very small K_λ for the case without finite ion size effects

We shall like to examine the asymptotic limits of Π_osm for very large and very small values of K_λ for the case where the Steric effect is neglected. Therefore, we start with eqn (13) and obtain:and

A.2. Derivation of the limiting values of osmotic pressure for very large K_λ for the case with finite ion size effects

Eqn (12) provides the expression for the osmotic pressure for the case with finite ion size effects. We estimate the limiting value of this osmotic pressure for large K_λ. For that purpose, we first estimate the limiting value of the term . One can write:

Using eqn (A3) in eqn (12), we can write:

We do not find such asymptotic limit for the case of K_λ≪1.

References

1. R. J. Hunter, Zeta Potential in Colloid Science, Academic Press, London, 1981.
2. J. Lyklema, Fundamentals of Interface and Colloid Science, Academic, San Diego, 1991.
3. H. Ohshima, Sci. Technol. Adv. Mater., 2009, 10, 063001.
4. H. Ohshima, Soft Matter, 2012, 8, 3511.
5. A. C. Barbati and B. J. Kirby, Soft Matter, 2012, 8, 10598.
6. H. Ohshima, Theory of Colloid and Interfacial Electric Phenomena, Elsevier/Academic, New York, 2006.
7. J. F. L. Duval and F. Gaboriaud, Curr. Opin. Colloid Interface Sci., 2010, 15, 184.
8. J. F. L. Duval, J. Merlin and P. Anantha, Phys. Chem. Chem. Phys., 2011, 13, 1037.
9. K. Makino and H. Ohshima, Sci. Technol. Adv. Mater., 2011, 12, 023001.
10. J. F. L. Duval, H. J. Busscher, B. van de Belt-Gritter, H. C. van der Mei and W. Norde, Langmuir, 2005, 21, 11268.
11. J. Langlet, F. Gaboriaud, C. Gantzer and J. F. L. Duval, Biophys. J., 2008, 94, 3293.
12. A. T. Poortinga, R. Bos, W. Norde and H. J. Busscher, Surf. Sci. Rep., 2002, 47, 1.
13. A. J. de Kerchove and M. Elimelech, Langmuir, 2005, 21, 6462.
14. R. Sonohara, N. Muramatsu, H. Ohshima and T. Kondo, Biophys. Chem., 1995, 55, 273.
15. R. Bos, H. C. van der Mei and H. J. Busscher, Biophys. Chem., 1998, 74, 251.
16. S. Tsuneda, J. Jung, H. Hayashi, H. Aikawa, A. Hirata and H. Sasaki, Colloids Surf., B, 2003, 29, 181.
17. K. D. Tachev, K. D. Danov and P. A. Kralchevsky, Colloids Surf., B, 2004, 34, 123.
18. S. S. Dukhin, R. Zimmermann and C. Werner, J. Colloid Interface Sci., 2004, 274, 309.
19. R. Zimmermann, S. S. Dukhin, C. Werner and J. F. L. Duval, Curr. Opin. Colloid Interface Sci., 2013, 18, 83.
20. J. F. L. Duval, R. Zimmermann, A. L. Cordeiro, N. Rein and C. Werner, Langmuir, 2009, 25, 10691.
21. J. F. L. Duval, D. Kuttner, C. Werner and R. Zimmermann, Langmuir, 2011, 27, 10739.
22. S. Chanda, S. Sinha and S. Das, Soft Matter, 2014, 10, 7558.
23. S. Das, Colloids Surf., A, 2014, 462, 69.
24. K. Makino, H. Ohshima and T. Kondo, Colloid Polym. Sci., 1987, 265, 911.
25. H. Ohshima, Colloid Polym. Sci., 2014, 292, 1227.
26. H. Ohshima, Colloid Polym. Sci., 2014, 292, 431.
27. H. Ohshima, J. Colloid Interface Sci., 2008, 328, 3.
28. H. Ohshima, J. Colloid Interface Sci., 2009, 333, 202.
29. H. Ohshima, J. Colloid Interface Sci., 2010, 350, 249.
30. H. Ohshima, Colloids Surf., A, 2011, 379, 18.
31. D. Ben-Yaakov, D. Andelman, D. Harries and R. Podgornik, J. Phys. Chem. B, 2009, 113, 6001.
32. R. P. Misra, S. Das and S. K. Mitra, J. Chem. Phys., 2013, 138, 114703.
33. M. S. Kilic, M. Z. Bazant and A. Ajdari, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75, 021502.
34. S. Chanda and S. Das, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2014, 89, 012307.
35. S. Das and S. Chakraborty, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 84, 012501.
36. S. Das, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85, 012502.
37. H. ShamsiJazeyi, C. A. Miller, M. S. Wong, J. M. Tour and R. Verduzco, J. Appl. Polym. Sci., 2014, 131, 40576.
38. N. Saleh, T. Sarbu, K. Sirk, G. V. Lowry, K. Matyjaszewski and R. D. Tilton, Langmuir, 2005, 21, 9873.
39. M. Okada, H. Maeda, S. Fujii, Y. Nakamura and T. Furuzono, Langmuir, 2012, 28, 9405.
40. S. Fujii, P. D. Iddon, A. J. Ryan and S. P. Armes, Langmuir, 2006, 22, 7512.
41. S. Fujii, K. Akiyama, S. Nakayama, S. Hamasaki, S. Yusa and Y. Nakamura, Soft Matter, 2015, 11, 572.
42. E. Boanini, P. Torricelli, M. C. Cassani, G. A. Gentilomi, B. Ballarin, K. Rubini, F. Bonvicini and A. Bigi, RSC Adv., 2014, 4, 645.
43. T. P. Bigioni, X.-M. Lin, T. T. Nguyen, E. I. Corwin, T. A. Witten and H. M. Jaeger, Nat. Mater., 2006, 5, 265.
44. G. Liu, M. Cai, X. Wang, F. Zhou and W. Liu, ACS Appl. Mater. Interfaces, 2014, 6, 11625.
45. J. Voigt, J. Christensen and V. P. Shastri, Proc. Natl. Acad. Sci., 2014, 111, 2942.
46. B. D. Storey and M. Z. Bazant, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 056303.
47. Y. B. Zhulina, V. A. Pryamitsyn and O. V. Borisov, Pol. Sci. USSR, 1989, 31, 205.
48. J. J. Lopez-Garcia, J. Hornoa and C. Grosse, J. Colloid Interface Sci., 2003, 268, 371.
49. J.-L. Liu and B. Eisenberg, J. Chem. Phys., 2014, 141, 22D532.
50. M. Z. Bazant, M. S. Kilic, B. D. Storey and A. Ajdari, Adv. Colloid Interface Sci., 2009, 152, 48.
51. S. Das, A. Guha and S. K. Mitra, Anal. Chim. Acta, 2013, 804, 169.
52. S. S. Dukhin, R. Zimmermann and C. Werner, J. Colloid Interface Sci., 2005, 286, 761.
53. S. Misra, S. Varanasi and P. P. Varanasi, Macromolecules, 1989, 22, 4173.
54. P. Pincus, Macromolecules, 1991, 24, 2912.
55. R. Israels, F. A. M. Leermakers, G. J. Fleer and E. B. Zhulina, Macromolecules, 1994, 27, 3249.
56. O. V. Borisov, E. B. Zhulina and T. M. Birshtein, Macromolecules, 1994, 27, 4795.
57. E. B. Zhulina and O. B. Borisov, J. Chem. Phys., 1997, 107, 5952.
58. R. R. Netz and D. Andelman, Phys. Rep., 2003, 380, 1.
59. E. Tsori, D. Andelman and J.-F. Joanny, Europhys. Lett., 2008, 82, 46001.
60. E. Yariv and A. M. J. Davis, Phys. Fluids, 2010, 22, 052006.
61. A. D. Phan, D. A. Tracy, T. L. H. Nguyen, N. A. Viet, T.-L. Phan and T. H. Nguyen, J. Chem. Phys., 2013, 139, 244908.
62. H. Ohshima, J. Colloid Interface Sci., 2003, 260, 339.

---