Article Cite This: Langmuir 2017, 33, 13133-13138
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Jammed Limit of Bijel Structure Formation P. M. Welch,* M. N. Lee, A. N. G. Parra-Vasquez, and C. F. Welch Theoretical Division and Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, United States S Supporting Information *
ABSTRACT: Over the past decade, methods to control microstructure in heterogeneous mixtures by arresting spinodal decomposition via the addition of colloidal particles have led to an entirely new class of bicontinuous materials known as bijels. Herein, we present a new model for the development of these materials that yields to both numerical and analytical evaluation. This model reveals that a single dimensionless parameter that captures both chemical and environmental variables dictates the dynamics and ultimate structure formed in bijels. We demonstrate that this parameter must fall within a fixed range in order for jamming to occur during spinodal decomposition, as well as show that known experimental trends for the characteristic domain sizes and time scales for formation are recovered by this model.
droplets containing bijels13 and in the form of particles, fibers, and membranes produced by solvent-transfer-induced phase separation (STRIPS).10,11 Polymers have also served as both the jamming agent and the immiscible fluid, as reported in the works of Virgilio18 et al. and Cui19 et al. The works of Lee,12,20,21 Mohraz,7 and their co-workers have illustrated the use of polymerized bijels as templates for nanocasting. Designing new bijel systems remains challenging because many of the basic principles dictating their formation are not fully understood. The need to more completely elucidate the guiding principles for constructing these materials has driven several theoretical8,22−27 studies beyond the pioneering work by Stratford.1 Some of the key properties targeted by these studies are the characteristic widths of the channels D, the impact of particle radius Ro, the time required for jamming to occur τ, and the combination of formulation variables needed for bijel formation. Since the early work by Herzig,3 many have argued that D ∝ 1/ρc, where ρc is the volume fraction of added particles. This has been found to be consistent with the Landau−Ginzburg model due to Hore and Laradji.24 However, even in the seminal work by Herzig,3 some data at lower volume fractions must be ignored for this trend to hold. Witt6 and co-workers argued that at low levels of particle inclusion, gravity destroys the bijel structure. This motivated their design of bridged bijels produced by introducing a slight preference for one of the two components of the phase-separating fluid, thereby stabilizing droplets formed by secondary nucleation and providing structural reinforcement for large liquid channels.
1. INTRODUCTION Bicontinuous interfacially jammed emulsion gels, bijels, emerged as a new route for producing materials with controlled bicontinuous structure roughly a decade ago, following their predicted existence from lattice Boltzmann simulations carried out by Stratford et al.1 First realized experimentally by Clegg2,3 and co-workers, these materials are typically produced by dispersing colloidal particles in binary fluids and subsequently thermally quenching the mixture into the unstable portion of the phase diagram, resulting in spinodal decomposition. The particles are chosen to preferentially partition to the liquid− liquid interface, where they are believed to jam and kinetically arrest the phase-separation process.4,5 This leads to multiphase materials with locked-in channel sizes on the order of 100 μm.6 As discussed by Witt7 and co-workers, porous materials derived from bijels present several attractive structural properties, including a uniform and interconnected pore geometry, a tunable interfacial surface area, and the ability to introduce active materials with a given thickness at the interface. These properties have led several investigators to suggest novel applications for these materials, including tissue engineering,8,9 catalysis,9 microfluidics,8,9 electrochemical devices,8 and filtration membranes.10 These potential applications motivated several synthetic approaches to producing bijels, with many studies pursuing advanced materials by using these systems as polymerization templates. Among the different fabrication schemes reported thus far are systems built from water, lutidine, and silica;3,12 nitromethane, ethylene glycol, and silica;13−15 edible bijels produced using spherical yeast and rodlike bacteria as the jamming agents;16,17 and styrene trimers, low-molecular-weight polybutene, and silica.8 Hierarchical structures constructed from bijels have also been produced in the form of Pickering © 2017 American Chemical Society
Received: August 25, 2017 Revised: October 13, 2017 Published: November 2, 2017 13133
DOI: 10.1021/acs.langmuir.7b02805 Langmuir 2017, 33, 13133−13138
Article
Langmuir
that the particle number fraction is given by ρ(R) = ∑Ni=1 δ(R − r) to write f/kT as eq 2.
Bai8 and co-workers fit their dynamical data for D to a simple empirically obtained exponential function. Reeves25 and coworkers argued that τ depends on the final channel thickness and final packing fraction for the particles, but the impact of the formulation variables remains unclear. The phase boundary between Pickering emulsions and bijels was probed by Jansen and Harting26 using a combination of molecular dynamics and lattice Boltzmann simulations. But again, we lack a clear criterion for determining whether a specific set of formulation variables result in the formation of the jammed state. Recently, Carmack and Millett27 demonstrated that a hybrid of a Cahn−Hillard−Cook model for the phase-separating fluid and a Brownian dynamics model for the particles faithfully reproduced much of the phenomenology of bijel formation. Herein, we analytically probe a similar but simpler model. Using a self-consistency argument, we demonstrate that the full data range for D found in Herzig’s3 original work may be fit, that an exponential function relates τ to Ro and other formulation variables, and that there exists a dimensionless parameter composed of the system’s variables that dictates the conditions needed for the formation of a bijel.
f a b κ c = ψ 2 + ψ 4 + |∇ψ |2 + ψ 2ρ kT 2 4 2 2
(2)
An equation of motion for the field ψ may be written and coupled to a Brownian dynamics algorithm for the evolution of the particles in the time-dependent field. The details of this are discussed in the Supporting Information. Employing this scheme, one sees that the salient features of bijel formation are recovered from the model encapsulated in eq 2. Figure 1
2. THEORY Consider a collection of monodisperse spherical particles moving in a background binary fluid. We define the supporting fluid in the usual Landau−Ginzburg28 formulation for an incompressible, symmetric binary solution. Each of the two components, A and B, are present in volume fractions ϕA and ϕB, respectively, such that ϕ0 ≡ ϕA + ϕB. The order parameter capturing the extent of phase separation is defined as ψ ≡ ϕA − 1 /2. The factor of 1/2 corresponds to the mean field critical volume fraction for such a system. The particles interact with the order parameter field via a simple harmonic potential. Thus, the component of the free-energy density f of the fluid associated with the phase-separating background (but not the particle−particle interactions) is given by eq 1. N
f κ a b c = ψ 2 + ψ4 + ∇ψ 2 + ∑ ψ 2δ(R i − r ) kT 2 4 2 2 i=1 (1)
Figure 1. Simulation snapshots from two numerical realizations of the combined Brownian dynamics and Landau−Ginzburg model. On the left, the system evolves without coupling the particles. On the right, 2400 particles are coupled to the phase-separating fluid field. Here, a = −20, b = 20, κ = 20, c = 5, and Ro = 0.25. Time is in arbitrary units.
The first two terms represent the drive of the background fluid toward phase separation when a is negative. The third term captures the penalty for forming an interface and reduces the curvature of those interfaces. The fourth term couples the particles to the field. We have added the fourth term in this form to address two issues: (i) We are formulating a field theory that has coarsened the description of the fluids to a continuum, but the particles are much larger than the fluid molecules. This difference in length scale leads us to couple a particle description to a background field when simulating the dynamics of bijel formation. Though not the focus of this study, we also aim to provide a new computational scheme for studying this class of material by coupling this model to a standard Brownian dynamics algorithm. (ii) A key experimental parameter in the formulation of these systems is the wettability of the particles by the fluids. By explicitly including the particle−field interactions rather than simply rescaling the interfacial penalty term κ, we provide an additional experimental connection for those who may wish to use the theory. This is similar in spirit to the formulation of the Helmholz free energy for particle adsorption put forward by Lin.29 The position of particle i is given by Ri. Next, we note
contains a series of snapshots from a two-dimensional simulation of this system both with and without coupled particles. Note that the interface quickly becomes established and the particles partition into this region in the images on the right with coupled particles. We now proceed to construct an analytical self-consistent evaluation of the density of particles at the interface in the jammed state, ρ*. Though not of direct interest, it serves as the link to predictions for several experimentally interesting quantities. For clarity of exposition, we first develop the theory in two dimensions and then generalize to three. As with the Landau−Ginzburg model for simple phase-separating blends, a characteristic length ζ may be determined by analyzing eq 2. Grouping the coefficients of ψ2, one finds that ζ = (κ/(|a| − cρ*)) 1/2 under the assumption that the particle density remains constant across the interface. This quantity is proportional to the interfacial thickness between the two phases.30 In the limit 13134
DOI: 10.1021/acs.langmuir.7b02805 Langmuir 2017, 33, 13133−13138
Article
Langmuir that the coupling between the phase-separating fluid and the particles is zero (viz., c → 0), one recovers the usual form for ζ in simple binary blends, ζ0. However, here one finds an additional term due to the coupling that is proportional to the density of particles ρ*. Again, we are considering the regime in which all of the particles are partitioned into the interfacial domain and ρ* refers to that local density; that is, we focus herein on the jammed state. We also note that, in effect, the fourth term in eq 1 rescales this interfacial thickness; we have herein elaborated on how the particles impact this quantity by explicitly including the particles and their coupling to the fluids. In building this model, we imagine that the phase separation process removes all available space for the particles except that in which they have been closely packed. Now, because ρ* is the number per unit area of particles within the interfacial region, we may estimate its value by assuming that all of the particles fall within an area (in two dimensions) given by ζL. Here, L is the total contour length of the interface in the entire fluid. Thus, ρ* is given by eq 3. 1/2 N N ⎛ |a| − cρ* ⎞ ⎟ = ⎜ ρ* = ⎠ L⎝ ζL κ
ρ* =
⎛ ⎞1/2 κ ζ=⎜ ⎟ ⎝ |a| − cρ* ⎠
⎛ ⎜ ζ = ⎜⎜ ⎛ ζ0 ⎜ 1 − ⎜χ 1/2 ⎝ ⎝
(3)
(4)
(5)
Now, we note that N/L ∝ 1/Ro, the inverse of the particle radius; that is, packing the particles in the interfacial region traces out a stack of circles that runs the full contour and L grows linearly with N. Given this, we may rewrite eq 5 by introducing a proportionality coefficient that captures this packing, g, as presented in eq 6. ρ* =
g R oκ 1/2
⎛ c 2g 2 ⎞1/2 cg 2 ⎜ ⎟ − a − 2 2R o 2κ ⎝ 4R o κ ⎠
⎛ ⎜ ⎜ =⎜ ⎡ ⎜ |a|⎢1 − ⎛⎜χ 1/2 ⎜ ⎝ ⎝ ⎣
1
(
χ 4
1/2
)
+1
κ
(
χ 4
1/2
)
+1
⎞1/2 ⎟ ⎟ ⎞⎤ ⎟⎟ − χ ⎟⎥ ⎟ ⎠⎦ ⎠
⎞1/2 ⎟ ⎞1/2 1 ⎟ ≡⎛ ⎜ ⎟ ⎞⎟ ⎝ 1 − F[χ ] ⎠ − χ⎟ ⎟ ⎠⎠
(9)
(10)
One may suspect that a divergence is lurking in eq 10 at the point where the function F[χ] becomes equal to 1. Because we are analyzing the jammed state, such a divergence would indicate an inconsistency in the theory. Careful inspection of that function reveals that it is always bounded to be less than approximately 0.2679 (details in the Supporting Information); no divergence exists in the model for jammed systems. This function does, however, allow one to identify limiting boundaries for the formation of the jammed state. Obviously, one can have only a jammed system when ρ* is greater than zero. Yet, it does fall to zero when F[χ] = 0, and this occurs at two different values of χ. When χ = 0, we have the trivial case of no coupling between the fluid and the particles (e.g., c = 0). 1 However, it also occurs at χ = 1 3 . Indeed, the ratio ζ/ζ0 goes through a maximum, beginning and ending at a value of 1 in the 1 range of χ = 0 to χ = 1 3 , as illustrated in Figure 2. Note that, though we have formulated this theory from the perspective of forcing the particles to the interface by driving them toward locations with mixed fluids, the resulting phenomenology is the
Solving for ρ* and taking only the physically relevant solution (that with positive ρ*) leads to eq 5. ⎞1/2 N ⎛ c 2N 2 cN 2 a − − ρ* = ⎟ ⎜ 2 ⎠ 2L2κ Lκ 1/2 ⎝ 4L κ
(8)
The interfacial thickness of the bijels is of direct interest because many envisioned applications of these materials rely upon the placement of functional particles at the interface and tuning the packing of the domain walls (e.g., monolayers, etc.). We may use eq 8 to relate the interfacial thickness in the bijel to that of the simple binary fluid blend ζ0, as given by eqs 9 and 10.
Rearranging eq 3, one obtains a quadratic expression for ρ*, given by eq 4. cN 2 aN 2 ρ*2 + 2 ρ* + 2 = 0 Lκ Lκ
⎡ ⎤ ⎞1/2 |a| ⎢ 1/2 ⎛ χ ⎜ χ + 1⎟ − χ⎥ ⎝4 ⎠ ⎥⎦ c ⎢⎣
(6)
Note that the assumed fixed proportionality of N/L is consistent with the physical notion that the density of particles in the interfacial region does not depend on the total number of particles in the system. Given that the interfacial region initially far exceeds the space taken up by the particles, this should be expected. The particles assume some concentration at the interfaces and lock a fraction of this boundary between the phase-separating fluids. By inspecting eq 6, one notes that a characteristic dimensionless ratio of system parameters dictates the behavior of ρ*. We define this ratio in two dimensions in eq 7. χ≡
c 2g 2 R o 2κ |a|
Figure 2. Ratio of the interfacial width with particles ζ divided by the corresponding width without coupled particles ζ0 as a function of the system’s dimensionless parameter χ. The values are obtained numerically from eq 10.
(7)
Thus, we may rewrite our expression for ρ* in a simpler form, given by eq 8. 13135
DOI: 10.1021/acs.langmuir.7b02805 Langmuir 2017, 33, 13133−13138
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Langmuir
dΔ = γνt γ− 1 dt
same as if one were formulating a theory to minimize the interfacial contact between two pure fluid phases. The interfacial width is generally larger when the particles are added, thus increasing the separation between the two pure phases. Therefore, we may write an approximate condition for the formation of the jammed state in two dimensions (eq 11).
Δ* ≡
c 2g 2
1 0<
