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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers
Diffusiophoresis of a Highly Charged Porous Particle Induced by Diffusion Potential Shan-Chi Tsai, and Eric Lee Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b04146 • Publication Date (Web): 04 Feb 2019 Downloaded from http://pubs.acs.org on February 8, 2019
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Langmuir
Diffusiophoresis of a Highly Charged Porous Particle Induced by Diffusion Potential Shan-Chi Tsai† and Eric Lee*,†
†Department
of Chemical Engineering
National Taiwan University No.1, Sec. 4, Roosevelt Road, Taipei, 10617 Taiwan Tel: +886-2-23622530
Keywords: Diffusiophoresis, Charged porous particle, Diffusion potential, Permeability, Double layer polarization effect, Counterion condensation effect
*:Corresponding author: Professor Eric Lee
E-mail:
[email protected] ACS Paragon Plus 1 Environment
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Abstract
Diffusiophoresis, the motion of a colloidal particle in response to the concentration gradient of solutes in the suspending medium, is investigated theoretically based on numerical computations in this study for charged porous particles, especially highly or extremely porous ones, focusing on the electrophoresis component induced by diffusion potential, which is generated spontaneously in a binary electrolyte solution where the diffusivities of the two ionic species are distinct. A benchmark carbonic acid solution of H+(aq) and HCO3－(aq) is chosen to be the major suspending medium, as its large diffusion potential and remarkable performance in practical applications have been reported recently in the literature. More than three orders of magnitude increase in particle diffusiophoretic mobility is predicted under some circumstances, should the permeability of the particle increases by ten-fold.
Nonlinear effects such as the motion-deterring double layer polarization effect pertinent to highly charged particles, and the counterion condensation or shielding/ screening effect pertinent to porous particles are investigated in particular for their impact upon the particle motion, among other electrokinetic parameters examined. A visual demonstration of the nonlinear double layer polarization is provided. Moreover, both the chemiphoresis and the electrophoresis components are explored and analyzed in detail.
The results presented here can be applied in biochemical and biomedical fields involving DNAs and proteins, which can be modelled excellently as charged porous particles in their electrokinetic motion.
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Introduction
Diffusiophoresis refers to the particle motion in response to a concentration gradient of solutes, electrolytes or non-electrolytes, in a suspending medium 1-4. Generally speaking, the imbalance of the interaction between the particle and the surrounding solutes provides the driving force which sets the particle in motion. This imbalance is resulted in general from an imposed concentration gradient upon the colloidal system 5-6. In a binary electrolyte solution, for example, the cations and anions will migrate in response to an imposed concentration gradient in the bulk solution by diffusion, according to the Fick’s law 7. If the diffusivities are different between the cations and anions, an electric potential is established spontaneously across the entire colloidal system due to the electro-neutrality constraint as no external electric field is applied. This induced potential, often referred to as the diffusion potential, tends to hinder the migration of faster ions (larger diffusivity) and enhance the migration of the slower ions (smaller diffusivity). The direction of this electric potential is thus determined. Once the system reaches steady state, the cations and anions will migrate at the same speed in the bulk solution due to the involvement of this diffusion potential. Note that this is the direct and straightforward consequence of the fundamental electrostatic Coulomb’s law between migrating ions. The presence of this diffusion potential is like a built-in “micro battery” 8 which drives the particle in motion following exactly the same mechanism as the conventional electrophoresis, hence is referred to as simply “electrophoresis” by some researchers to highlight the similarity between them 2. Actually diffusiophoresis is one family member of the electrokinetic motions. Note that this “electrophoresis by diffusion potential” is closely related to a dimensionless factor β, defined as β
D1 D 2 for a symmetric binary electrolyte solution D1 D 2
, where D1 and D2 are diffusivity coefficients of cations and anions, respectively 2. Note that direct measurements of diffusion potential for various electrolyte solutions are reported as well 9,
in addition to the theoretical calculation of β by definition here.
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The other mechanism is due to the screening on the borderline of the double layer, which favors the migration of counterions whereas rejects the entry of coions due to the Coulomb attractive or repulsive forces between the charged particle and the electrolyte ions surrounding it. This mechanism is often referred to as the chemiphoresis component with the implication that it is induced by the difference of chemical potential (or concentration) of solutes within the double layer, hence can be viewed as the consequence of the double layer polarization or redistribution
10.
Note that the chemiphoresis phenomenon in diffusiophoresis is
fundamentally different from the double layer polarization in the conventional electrophoresis. There is a net ion flux across the double layer in chemiphoresis, whereas the double layer polarization in conventional electrophoresis is induced by an superimposed external electric field upon the entire double layer simultaneously, which leads to a redistribution of counterions within the double layer. There is no net flow of ions across the boundary of the double layer in conventional electrophoresis. This fundamental difference in the nature of double layer polarization leads to completely different behaviors of particle motion. Once the particle is in motion, either by chemiphoresis alone or with the involvement of additional electrophoresis induced by diffusion potential, it will be balanced instantly by the hydrodynamic drag force following the famous Stokes law 11.
Diffusiophoresis of rigid particles has been extensively investigated in various theoretical aspects
2, 12-22.
For a rigid particle with very thin double layer, a simple direct proportional
relationship is available as follows in a symmetric binary electrolyte solution 8, 23:
U dp
εkT kT zeζ C βζ 2 ln(1 tanh 2 ( )) ηze ze 4kΤ C
(1)
where Udp is the particle diffusiophoretic velocity, C is the solute concentration in the bulk electrolyte solution, and ζ is the zeta potential of the particle. The meaning of the rest of the symbols can be found in the List of Symbols. The first term stands for the electrophoresis ACS Paragon Plus 4 Environment
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Langmuir
component, and is proportional to the diffusion potential in this case. The second term is the chemiphoresis component originating from the double layer polarization favoring the counterions, as mentioned earlier. And β is the β factor defined earlier. Note that the particle speed is proportional to ∇C/C, instead of ∇C alone. Hence a small concentration gradient may still generate significant particle motion in a dilute electrolyte solution. This is a unique feature pertinent to diffusiophoresis and has also been observed experimentally for rigid particles 24-25. Note that an analytical formula for chemiphoresis is also available for arbitrary double layer thickness under the Debye-Hückel approximation 26-27, in other words, when the charge level of the particle is low.
According to Eq. (1), searching for an electrolyte solution yielding large β is essential for the enhancement of diffusiophoretic motion to gear up the overall process rate in potential practical applications. Normally NaCl is chosen as the benchmark suspending medium in academic studies as it is probably the most frequently encountered electrolyte on earth and human body, in the form of the sea water and human blood. Its β factor, however, is only about - 0.2, and more often than not it is not sufficient to induce fast enough diffusiophoretic motion of colloidal particles for practical applications. Note also that the exact β value is -0.03 for KCl, which is often regarded approximately as zero in practice, such as in the classic work by O’Brien and White28 in electrophoresis.
For finite double layer thickness, the motion-deterring double layer polarization effect in general has to be considered for the prediction of the particle velocity
13-20.
The governing
electrokinetic equation becomes highly nonlinear and coupled hence no analytical formula is available any more for highly charged rigid particles as well as other highly charged colloidal entities, such as the porous particle, which is an excellent model for polyelectrolytes like DNAs and proteins in general. Moreover, the electrophoresis and chemiphoresis mechanisms can no longer be separated and numerical solution is needed in general. Lee’s group in particular has conducted many studies in this field regarding the double layer polarization ACS Paragon Plus 5 Environment
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effect
14-20.
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Double layer polarization refers to the uneven distribution of ions in the double
layer when the particle is in motion. The predominant counterions in the double layer migrate toward the opposite direction of the moving particle due to the Coulomb’s electrostatic law for one thing, and the convection fluid flow around the moving particle will further enhance this motion-deterring double layer polarization effect by bringing forth more counterions in the double layer to the wake of the moving particle. The polarization/deformation of the double layer is anti-symmetric in nature, and manifests its effect when the double layer thickness is comparable to the particle radius.
Diffusiophoresis has found many novel applications in recent years. It has been used as a micro-pump with self-generated diffusiophoretic flows from colloidal calcium carbonate 29, for instance. Detection and remediation of bone crack has also been proposed using the concentration gradient of calcium electrolyte ions (Ca+) in the vicinity of the injury area to guide the drug-carrying quantum dot particles reaching there. Other applications are also encouraging, such as the extraction of cells from dead-end pores in various microcavity environments 5, extraction of oil droplets from porous oil-containing rocks in the enhanced oil recovery (EOR) operation
30-31,
enhancement of washing performance by establishing a
surfactant concentration gradient to drive contaminants from pores 32. Indeed diffusiophoresis has evolved from the traditional role of an esoteric laboratory phenomenon to embrace the brane new world due to the unique feature that other candidates can not match 8, such as there is no Joule heating effect in diffusiophoresis, which is fatally detrimental to mammalian cells if the temperature is raised above 40 ºC 33, or 3 ºC higher than the normal body temperature at about 37ºC. Moreover, diffusiophoresis also found enormous potential applications in biomedical and biochemical fields in the form of Janus particles, where asymmetric layout of coated catalyst provides the driving force from the imbalance of the osmosis pressure around the particle 34-35
In addition to the various novel applications mentioned above, the membraneless water ACS Paragon Plus 6 Environment
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filtration using CO2 gas is very interesting in particular 36. Nearly 92 mV self-generated electric potential can be established across an aqueous electrolyte solution of carbonic acid H2CO3 by the introduction of CO2 gas into water, which generates cations H+ and anions HCO3－. Strong diffusiophoretic motion of particles are demonstrated experimentally which greatly enhances the purification performance on polluted water. The diffusiophoresis mechanism proposed there has various merits such as low energy consumption and no membrane-fouling problem 36.
The demonstration particles used there are the rigid polystyrene latex particles (PS). In the
general systems encountered in the wastewater treatment, however, very often highly or even extremely permeable pollutants or contaminants are encountered as well. Moreover, as mentioned earlier, the charged porous particle is also an excellent model to describe the electrokinetic behavior of polyelectrolytes such as proteins and DNAs. Hence, it is desirable to explore the diffusiophoretic behavior of highly or extremely permeable colloidal particles in electrolyte solution with large diffusion potential (large β) as well, such as the H2CO3 aqueous solution mentioned above, whose β value is as large as 0.774, in comparison with the commonly encountered NaCl solution, whose β value is only - 0.2. The results should provide more insights and extend the scope of the potential applications of diffusiophoresis to even broader fields, as the particle speed is in general much faster for a permeable one than a rigid one 20, 37-40.
We thus decide to conduct a rigorous theoretical analysis for a highly or even extremely permeable porous particle to investigate its diffusiophoretic behavior in this promising aqueous electrolyte solution of H2CO3. Note that the presented theory is not limited to the H2CO3 system chosen as the demonstration example here. Any other electrolyte solution with large β can be used as the suspending system as well. The effect of β factor and other electrokinetic parameters will be examined as well to shed light on this interesting system with promising potential in practical applications, such as the purification of polluted water, which is an invaluable resource in desert area in particular. Moreover, the modelling of proteins and DNAs with the current analysis here certainly can be applied directly in biomedical and ACS Paragon Plus 7 Environment
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biochemical fields for their diffusiophoretic motions there , as reported in some experimental works of protein diffusiophoresis 4, 41-42.
In summary, the chemiphoresis (β = 0) for a charged porous spherical particle has been investigated by Lee’s group recently
20.
Here we extend it further to consider the
electrophoresis component induced by diffusion potential, and focus on the highly or even extremely permeable situation in an electrolyte solution with strong diffusion potential, such as the aqueous H2CO3 solution 36. Note that the electrophoresis component induced by the internal diffusion potential is different from the pure electrophoresis of a porous particle subject to an external electric field, which has been investigated extensively by H. Ohshima under Debye-Hückel approximation
43-47.
With this study, thorough understanding of the
general diffusiophoresis behavior of the porous particles can be achieved, taking into account the contributions from both the chemiphoresis component (β = 0) and the electrophoresis component (β ≠ 0).
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Theory
The system diagram is shown in Fig. 1, where the diffusiophoretic motion of a spherical charged porous particle suspended in a binary z1:z2 electrolyte solution is considered. A concentration gradient of ions ∇C in the bulk solution is imposed upon the system in the upward z-direction, which drives the particle in motion with a constant velocity U along the z-direction.
The governing equations here are the general electrokinetic equations valid for all electrokinetic systems, with the diffusiophoresis considered here a member to it
48.
They
consist of the Gauss theorem, the modified Stokes equation with an extra term accounting for the local electric body force, the conservation of ions and the incompressibility constraint of the fluid flow:
2
2
fix , 0
