Subscriber access provided by Kaohsiung Medical University
Interfaces: Adsorption, Reactions, Films, Forces, Measurement Techniques, Charge Transfer, Electrochemistry, Electrocatalysis, Energy Production and Storage
Experimental and Theoretical Validation of System Variables That Control the Position of Particles at the Interface of Immiscible Liquids Sarah Innes-Gold, Christopher J Luby, and Charles R. Mace Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01197 • Publication Date (Web): 08 Jun 2018 Downloaded from http://pubs.acs.org on June 8, 2018
Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.
Page 1 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Experimental and Theoretical Validation of System Variables That Control the Position of Particles at the Interface of Immiscible Liquids
Sarah N. Innes-Gold, Christopher J. Luby, and Charles R. Mace*
Department of Chemistry, Tufts University, 62 Talbot Avenue, Medford, MA 02155 United States
*Corresponding author:
[email protected] ACS Paragon Plus Environment
1
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 2 of 28
Abstract We construct a mathematical model describing the equilibrium flotation height of a spherical particle at the interface of immiscible liquids. The behavior of such a system depends on several experimentally measurable parameters, which include surface tensions, densities of all phases, and system scale. These parameters can be absorbed into three quantities that entirely determine the equilibrium position of the particle: the contact angle between the interface and particle, the Bond number, and the ratio of particle buoyant density to liquid phase densities—a new, dimensionless number that we introduce here. This experimentally convenient treatment allows us to make predictions that apply generally to the large parameter space of interesting systems. We find the model is in good agreement with experiments for particle size and interfacial tension spanning three orders of magnitude. We also consider the low interfacial tension case of aqueous two-phase systems (ATPS) theoretically and experimentally. Such systems are more sensitive to changes in density than higher tension aqueous-organic twophase systems; we experimentally demonstrate that a millimeter-sized bead in an ATPS can be controllably positioned with between 5.9% and 95.1% of its surface area exposed to the bottom phase, while the same bead in an aqueous-organic system is limited to a range of 18.2–61.6%. Finally, we discuss the potential for wettability-based control for micron length-scale particles, which are not sensitive to changes in density. Our results can be used to simply define the experimentally controllable parameters that affect a particle’s equilibrium position and the length scales over which such parameters can be effectively tuned. A complete understanding of these properties is important for a number of applications including colloidal self-assembly and chemical patterning (e.g., formation of desymmetrized or Janus particles). By considering ATPS, we broaden the potential uses to biological applications such as cell separation and interfacial tissue assembly.
ACS Paragon Plus Environment
2
Page 3 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
Introduction Solutes introduced to phase separated systems prepared from immiscible liquids will often partition preferentially into a single phase, where the completeness of partitioning or enrichment is characterized by an equilibrium constant that considers the differences in concentrations of the solutes between phases. Insoluble objects (e.g., polymeric particles or cells) also experience partitioning behavior in two-phase systems based on their affinity for either phase. The forces driving this partitioning may be related to charge, hydrophobicity, and binding affinity between phase components and the surface of the object.1,2,3 However, upon application of an external centrifugal field, the equilibrium position of these objects will reside at an interface within the system—air/top phase (gas/liquid interface) if the object is less dense than the top phase liquid, top phase/bottom phase (liquid/liquid interface) if the object density is intermediate between the immiscible liquids, or bottom phase/container (liquid/solid interface), if the object is more dense than the bottom phase liquid.4,5,6 While sedimentation to a specific interface is driven by buoyancy, the absolute vertical position of an object at a liquid/liquid interface is controlled significantly by the surface tensions within the system (i.e., interfacial tension of the two-phase system and surface tension of the object). As a result, a potentially tunable fraction of the object’s surface area will be exposed exclusively to each bulk phase. Precise positioning of objects at liquid/liquid interfaces has a number of applications in chemical patterning of desymmetrized (i.e., Janus) particles.7 The asymmetry of Janus particles gives rise to individual behaviors and complex systems that are inaccessible to homogeneous particles.7 As a result, Janus particles have important applications in biology (e.g., as a vehicle for the simultaneous delivery of hydrophilic and hydrophobic drugs),8 electronics (e.g., charged and colored spheres that align under an electric field to produce a reconfigurable display)7,9 and self-assembly (e.g., programmable colloids). 10 , 11 Current approaches to synthesize such particles allow for limited control over the extent of the patterned area.12–21 Further emerging applications exploiting the localization or precise positioning of objects at liquid/liquid interfaces
ACS Paragon Plus Environment
3
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 4 of 28
use immiscible aqueous systems—aqueous two-phase systems (ATPS) or, more broadly, aqueous multiphase systems (AMPS)22—for the separation of cells by density5,6 or the directed assembly of cells into tissue-like constructs. 23 Both classes of applications are only made possible through the use of water as the common solvent for these phase-separated systems. The ability to precisely position or assemble particles and cells at liquid interfaces requires a detailed understanding of the forces acting on these particles and how the forces scale with particle size and system properties. This understanding allows for the properties of the system or the particle itself to be deliberately tuned in order to achieve experimental results and explain observed phenomena. In this manuscript, we implement a force-balance model to describe the position of a spherical particle at the interface of immiscible liquids based on the gravitational and interfacial forces acting on the particle at equilibrium. We focus on which experimentally controllable factors dominate in determining the vertical position of the particle, and as an example consider for what systems liquid phase density might be a useful control variable for positioning the particle at a chosen height. Our model predicts over what length- and tension-scales a full range of particle positions can be achieved by tuning density. We reframe the traditional force-balance treatment in terms of two dimensionless quantities calculated from easily measurable system parameters. Doing so allows the theory to make predictions that encompass a large range of experimentally interesting systems while reducing the number of parameters that must be specified independently. We experimentally validated the applicability of our model using a large range of particle sizes (micrometer to millimeter) and interfacial tensions (µN/m to mN/m). As a test of the model, we looked at large objects (several millimeters) in relatively high-tension systems (i.e., aqueousorganic) and found that changing the density of the system through the addition of solutes could dramatically tune the object position, and that the theory yielded accurate predictions. In low tension systems (i.e., ATPS), a nearly full range of positioning could be achieved for millimeter-
ACS Paragon Plus Environment
4
Page 5 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
sized objects using only small changes in density, but this extreme sensitivity to density leads to less accurate predictions. Our theory-based sensitivity analysis for the position of a bead at a two-phase liquid interface, validated versus experiments, is useful in determining for what experimental regimes the mathematical model is effective as a predictive tool, with applications in industrial processes, chemical modification of materials, and the separation and assembly of cells.
Experimental Section Theoretical Considerations We used a force balance approach to model the equilibrium position of a spherical bead at an interface (Figure 1). All calculations were performed using MATLAB. The system is described by bead radius R (m), bead density (kg/m3), top and bottom phase densities and (kg/m3), interfacial tension (N/m), interfacial deformation height d (m), equilibrium contact angle , and angle of inclination , which parameterizes the bead height at the interface. Densities, bead radius, interfacial tension, and equilibrium contact angle are the experimental quantities that must be known in order to solve for and obtain the equilibrium bead position. Three forces determine the bead’s vertical position: (i) gravity and (ii) buoyancy act along the vertical and (iii) interfacial tension acts tangentially to the interface in the direction determined by the contact angle . In terms of the previously defined system parameters, the expression for the gravitational force is
= − .
(1)
The buoyant force is
= 2 − 3 cos + cos + 2 + 3 cos − cos − − sin (2)
and the vertical component of the interfacial tension force is = −2 sin sin +
ACS Paragon Plus Environment
(3)
5
Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Page 6 of 28
(see Supporting Information for derivations). These expressions agree with previous treatments of the system described by others.24–27 At equilibrium, the forces defined in Eqn. 1–3 sum to zero: 0 = + +
(4)
The goal of the model is to solve Eqn. 4 for , the angle of inclination,28 which can be used to find the surface area below the interface "#$%&' = 2 − cos
(5)
The area calculated by Eqn. 5 is then divided by the total surface area to give the fraction of the particle area below the interface, which we use to characterize the vertical position of the particle in the system with respect to the liquid/liquid interface. Note that the buoyant force depends on d, the meniscus height where the interface meets the bead. Some formulations treat d as an experimentally determined system parameter.25,29 In fact, this height obeys the Young-Laplace equation for a meniscus and depends entirely on the other system parameters.24,30 The dependence of d on other parameters must be known in order to use Eqn. 4 for cases when the particle is held at the interface by interfacial tension.26 The Young-Laplace equation describing the interface profile around a spherical particle has no closed-form solution.28, 31 In order to complete the force-balance description of the system, we approximate the meniscus height using numerical methods based on existing algorithms (see Eqn. S12 in Supporting Information for details).24,32,33 It is convenient to introduce two dimensionless quantities: the dimensionless particle density D=
)* +), )- +),
,
(6)
B=
)- +), / 0 1 2
.
(7)
and the Bond number26
ACS Paragon Plus Environment
6
Page 7 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Langmuir
D accounts for the relative densities of all phases of the system, while B compares the scale of gravitational forces and interfacial tension forces. In a system of large Bond number (B >> 1) surface tension effects are negligible compared to gravity and buoyancy. If the Bond number is small (B
