Dynamics of Encapsulation and Controlled Release Systems Based

Oct 8, 2008 - Dynamics of Encapsulation and Controlled Release Systems Based on Water-in-Water Emulsions: ... The Journal of Physical Chemistry B...
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J. Phys. Chem. B 2008, 112, 13503–13508

13503

Dynamics of Encapsulation and Controlled Release Systems Based on Water-in-Water Emulsions: Negligible Surface Rheology Leonard M. C. Sagis* Physics Group, Department ATV, Wageningen UniVersity, Bomenweg 2, 6703 HD Wageningen, The Netherlands ReceiVed: July 8, 2008; ReVised Manuscript ReceiVed: August 19, 2008

A nonequilibrium thermodynamic model based on the interfacial transport phenomena (ITP) formalism was used to study deformation-relaxation behavior of water-in-water emulsions. The ITP formalism allows us to describe all water-in-water emulsions with one single theory. Phase-separated biopolymer solutions, hydrogel beads, liposomes, polymersomes, colloidosomes, and aqueous polymer microcapsules are all limiting cases of this general theory with respect to rheological behavior of the bulk phases and interfaces. Here we have studied two limiting cases of the general theory, with negligible surface rheology: phase-separated biopolymer solutions and hydrogel beads. We have determined the longest relaxation time for a small perturbation of the interfaces in these systems. Parameter maps were calculated which can be used to determine when surface tension, bending rigidity, permeability, and bulk viscoelasticity dominate the response of a droplet or gel bead. In phase-separated biopolymer solutions and dispersions of hydrogel beads six different scaling regimes can be identified for the relaxation time of a deformation. Hydrogel beads may also have a damped oscillatory response to a deformation. The results presented here provide new insight into the complex dynamics of water-in-water emulsions and also suggest new experiments that can be used to characterize the interfacial properties of these systems. Introduction In a recent review paper a general model was developed for the dynamic behavior of water-in-water emulsions.1 Water-inwater emulsions are dispersions of an aqueous phase in another aqueous phase, like for example phase-separated biopolymer solutions, and aqueous dispersions of liposomes, polymersomes, colloidodomes, hydrogel beads, or aqueous core-shell polymer microcapsules. These systems can be used for encapsulation and controlled release purposes in for example pharmaceutical, food, or cosmetic products. The stress-deformation behavior of droplets in water-in-water emulsion is highly complex and affected by mass transfer across the interfaces, predominantly of water, but also of dissolved species.1 For example, in phase-separated biopolymer solutions the relaxation of droplet shape in cessation of flow experiments is significantly accelerated by mass transfer.2,3 Mass transfer also affects the results of surface tension measurements in these systems with the spinning drop method.4 The stress-deformation behavior of colloidosomes is also affected by mass transfer.5 Microcantilever experiments show that the volume of colloidosomes is not conserved during deformation and that water is transferred from the interior of the colloidosomes to the continuous phase.5 The coupling between stress-deformation behavior and mass transfer of encapsulates was also observed in hydrogel beads, where the release of encapsulants from the beads was retarded by both shrinking and swelling of the particles.6 For the design of new encapsulation or controlled release systems based on water-in-water emulsions, a better understanding of the dynamic behavior of these emulsions is essential. The general theory developed in ref 1 takes into account the * E-mail: [email protected].

coupling of the stress-deformation behavior and mass transfer across the interfaces. This theory was developed using the interfacial transport phenomena (ITP) formalism, a nonequilibrium thermodynamics formalism for the description of dynamic behavior of multiphase systems, with excess parameters (surface tension, surface density, etc.) associated with the interfaces.7 The theory can be applied to all classes of waterin-water emulsions. In fact, phase-separated biopolymer solutions and aqueous dispersions of liposomes, polymersomes, colloidosomes, hydrogel beads, and polymer core-shell particles are basically limiting cases of the general theory, with respect to rheological behavior of the bulk phases and the interfaces. In this paper we will focus on two of these limiting cases: phase-separated biopolymer solutions and hydrogel beads. These systems have in common that the contribution from surface rheology (surface viscosity and surface elasticity) to the dynamic behavior of the droplets or beads is in general negligible. But they represent different limiting cases for the bulk rheological behavior. For the phase-separated biopolymer solutions the rheological behavior of the bulk phases is Newtonian, as long as the concentrations and molecular weight of the polymers are relatively low.2,3 For gel beads or phase-separated polymers at high concentration the behavior is linear viscoelastic. We will focus on the expression for the longest relaxation time of a perturbation of the interfaces of the droplets or beads. This relaxation time is not only a useful parameter to predict the behavior of a droplet in a flow field but can also be used to determine properties of the interfaces of droplets, using for example cessation of flow experiments.2,3 For the water-in-water emulsions studied here, the scaling of the relaxation time with droplet size and deformation amplitude will be discussed and compared to available experimental data. We will show that we can identify six different scaling regimes, depending on the relative magnitudes of the bending rigidity, surface tension,

10.1021/jp806014b CCC: $40.75  2008 American Chemical Society Published on Web 10/08/2008

13504 J. Phys. Chem. B, Vol. 112, No. 43, 2008

Sagis

(i) (i) (i) F(dex) F(w) F(gel) phase (kg/m3) (kg/m3) (kg/m3)

1 2

22.0 50.0

37.5 16.0

952.3 944.2

η(i) (Pa s)

ν(i) (Pa s)

γ0 (N/m)

k (kBT)

7.4 × 10-3 17.4 × 10-3 0.2 × 10-6 200 8.2 × 10-3 19.1 × 10-3

interfacial permeability, bulk and shear viscosities of the interior and continuous phase, and composition of these phases. A parameter map is constructed which can be used to determine when these parameters dominate the relaxation behavior of the system. Theory Governing Equations. The theory is a continuum theory based on the ITP formalism7 and starts with the differential balance equations of the system: the equations of continuity and differential momentum balances for the bulk phases and the jump mass balances and jump momentum balance for the interfaces.7 Here we will give only a brief summary of the theory. A complete description of the theory is presented in ref 1. The theory is valid for small deformations of the droplet interface in an N-component mixture and neglects any inertial effects in the momentum balances for the bulk phases and the interfaces. We will assume there is no imposed flow field and limit ourselves to a study of the thermal fluctuations of the interfaces. In addition, we will assume that the concentration fluctuations are sufficiently small, such that the effect of contributions from ordinary diffusion on the dynamic behavior can be neglected. For the surface tension γ of the interface the Helfrich expansion8 will be used, i.e., γ ) γ0 - kC0H + kH2 + jkK, where γ0 is the surface tension of the flat interface, k is the bending rigidity of the interface, C0 is the spontaneous curvature, H is the mean curvature of the interface, K is the Gaussian curvature, and jk is the rigidity constant associated with K. The coupling of the stress-deformation behavior and mass transfer across the interface is obtained by using the following constitutive equation for the mass flux vectors of species A, j(A) (A ) 1,..., N), evaluated at the interface: (1) (2) j) (j(A) - j(A) ) · ξ ) λp∆Fj(A)(∆P - ∆P

(

)

∂R(θ, φ, t) 2γ (1) (1) (2) (2) - (zj(A) Vr - jz(A) Vr ) ) 0 - λp P(1) - P(2) ∂t R0

TABLE 1: Densities and Viscosities for the GelatinDextran-Water System2,3

(1)

where ξ is the unit normal vector of the interface, λp is the (1) (2) (i) permeability of the interface, ∆Fj(A) ) Fj(A) - Fj(A) , Fj(A) is the (1) equilibrium density of species A in phase i, ∆P ) P - P(2), and P(i) is the pressure in phase i. Since the equilibrium dividing j ) 2γ/R0. In this surface is a spherical interface, we have ∆P particular form (1) predicts zero mass transfer across the interface at equilibrium and gives an increasing mass flux with increasing deformation of the interface. As mentioned above, in arriving at (1), we have assumed that the contributions of ordinary diffusion across the interface are negligible. This is (1) (2) j F(A)(∆P - ∆P j ), where k(A) valid when k(A)(δF(A) - δF(A) ) , λp∆ ) D(A)/l is the mass transfer coefficient for species A, D(A) is the diffusion coefficient of species A, and l is the thickness of the interface. Accounting for ordinary diffusion will introduce a second characteristic time that scales as τod ∼ 1/(q2D(A)) and is independent of the deformation of the droplet.9,10 It will not affect the expression for the longest relaxation times τBL and τHL calculated in the next sections. With the assumptions listed above, the time evolution of the parametrization of the interface, R(θ,φ,t), for small perturbations is given by1

(2) (i) (i) where (r, θ, φ) are spherical coordinates, jz(A) ) Fj(A) /∆Fj(A), and Vr(i) is the r-component of the mass-averaged velocity in phase i. For the jump momentum balances we find1

(divs σs)r + (γ0 - kC0R0-1)∇2s R(θ, φ, t) 1 2 2 (2) (1) - σrr ) 0 (3) k∇ ∇ R(θ, φ, t) + P(1) - P(2) + σrr 2 s s Here (divs σs)r denotes the r-component of the surface divergence of the surface extra stress tensor σs, ∇2s denotes the surface Laplacian, and σ(i) rr is the rr-component of the extra stress tensor of phase i. The linearized equations of continuity and linearized momentum balances for the bulk phases are given by

∂δF(i) + Fj(i)div v(i) ) 0 ∂t

(4)

∇P(i) ) div σ(i) + fh(i)

(5)

where δF(i) are the fluctuations of the total mass density, Fj(i) is the equilibrium total mass density, and fh(i) is the force resulting from thermal fluctuations. Equations 2-5 form a set of coupled equations that needs to be completed with constitutive equations for the extra stress tensors of the bulk phases and the surface extra stress tensor. For realistic constitutive models for these tensors and in an arbitrary imposed flow field, an analytical solution of this set of equations does not exist, and the equations have to be solved numerically. But for a number of limiting cases analytical solutions can be found. Two of these limiting case will be discussed in the following section. Rheological Behavior of Bulk Phases. For the phaseseparated biopolymer solutions the fluid phases on both sides of the interface are assumed to be Newtonian. The expression for the extra stress tensor is given by

σ(i) ) ν(i)(div v(i))δ + 2η(i)D(i)

(6)

(i) 2 (i) (i) Here ν(i) ) η(i) b - /3η , η is the shear viscosity, ηb is the (i) bulk viscosity, and D is the rate of deformation tensor. For the hydrogel beads the exterior phase is assumed to be Newtonian, and the extra stress tensor of the gelled phase is assumed to be given by an integral linear viscoelastic model11,12

σ(i) )

∫-∞t N(i)(t - t')[div v(i)(t')]δ dt' + t 2∫-∞ G(i)(t - t')D(i)(t')] dt' (7)

where G(i)(t) is the relaxation modulus of the gelled phase, N(i)(t) ) K(i)(t) - 2/3G(i)(t), and K(i)(t) is the bulk relaxation modulus of the gelled phase. When the rheological behavior of a material is described by eq 7, it is referred to as a fading memory fluid.11 Results and Discussion Phase-Separated Biopolymer Solutions. We now proceed with the calculation of the longest relaxation time for a phaseseparated biopolymer solution. A common procedure to determine the relaxation time is to expand R(θ,φ,t) and the velocity and pressure fields in spherical harmonics. To facilitate comparison of our results with previous results obtained for flat interfaces in a phase-separated biopolymer solution,9,10 we will not use an expansion in spherical harmonics, but instead Fourier transform the set of equations. As a result, we will obtain a

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continuous spectrum of relaxation times, rather than a discrete one, and need to introduce a realistic cutoff to obtain an expression for the longest relaxation time. We first transform (2), then eliminate the velocity fields using the Fourier transform of (4) and (5), and eliminate the pressure term using the transform of (3). We find that the q-dependent relaxation time τ(A)B is given by (see Appendix A)

τ(A)B )

2 -1 |

(γ + 21 kq )

1 q -2 (1) (2) | λp + φ(A) + φ(A) (i) φ(A) )

0

(i) Fj(A)

(8)

(9)

∆Fj(A)κ(i)qr

where q| is the magnitude of the component of the wave vector q parallel to the interface, qr is the r-component of q, and κ(i) 4 (i) ) η(i) b + /3η . In arriving at this result, we neglected the contribution from the term containing the spontaneous curvature, so we assumed that 2kC0R0-1 , 2γ0 + kq|2. We expect the effects of spontaneous curvature to be important only in liposomes or polymersomes. Equation 8 is a generalization of the expression derived in Prost et al.13 for the relaxation time of a flat membrane and the semiempirical expression used in Scholten et al.2,3 From eq 8 we can calculate an expression for the longest relaxation time of the interface, τ(A)BL. For a spherical interface the longest wavelength for a perturbation is proportional to R0. The corresponding value for q| is 1/R0. If we denote the amplitude of the perturbation by δR0, then qr ∼ 1/δR0. Substituting this in (8), we find

τ(A)BL )

2R04

1

(1) (2) λp + φ(A) + φ(A) 2γ0R02 + k (i) φ(A) )

(i) Fj(A) δR0

(10)

(11)

∆Fj(A)κ(i)

Equation 28 not only gives us the relaxation time but can also be used to calculate the decay of the correlation function of the fluctuations of the interface.9,10 In the long-time limit (ω f 0) we find that the decay of this function is given by10 N

〈δR(q, t)δR*(q, t) 〉 ∼

∑ e-t ⁄ τ

(A)B

(12)

A)1

We see that, depending on the permeability of the interfaces, composition and viscosities of the bulk phases, and amplitude of the perturbation, each component in the mixture may have a distinct contribution to the decay of the correlation function. This multiexponential response to a deformation was also observed by Scholten et al.4 in spinning drop experiments on a gelatin-dextran-water system. Equation 12 can be used to extract data for the permeability, bending rigidity, and surface tension, from the shape fluctuations of droplets or hydrogel beads, measured by for example neutron spin echo experiments14,15 or diffusing-wave spectroscopy.16,17 Equation 10 can be used to investigate the scaling behavior of τ(A)BL with R0 and δR0. For sufficiently small deformations, (1) (2) (2) (1) such that δR0 , λp∆Fj(A)/B, where B ) (F(A) κ + F(A) κ )/ (1) (2) (κ κ ), we can distinguish two different scaling behaviors for the relaxation time τ(A)BL. For small droplets, the relaxation time is dominated by the bending rigidity, and τ(A)BL scales as

τ(A)BL ∼ R04 ⁄ (λpk)

Figure 1. Scaling of the relaxation time τ(A)BL with droplet radius R0, for the sample in Table 1, at a deformation amplitude δR0 ) 1.0 × 10-8 m, and three values of the permeability λp: (4) λp ) 1.0 × 10-4 m3/(N s), (]) λp ) 3.4 × 10-3 m3/(N s), and (0) λp ) 5.0 × 10-2 m3/(N s).

(13)

This scaling is in agreement with the scaling found for rigidity and permeability dominated coarsening during phase separation

Figure 2. Scaling of the relaxation time τ(A)BL with droplet radius R0, for the sample in Table 1, at a deformation amplitude δR0 ) 2.0 × 10-5 m: The solid line was calculated with eq 10, choosing λp ) 5.5 × 10-3 m3/(N s), and 0 denotes experimental data from Scholten et al.2,3 The dashed line is the scaling τ ∼ R0η/γ0, valid for impermeable interfaces.18

in gelatin-dextran-water systems.3 For large droplets, the relaxation time is almost insensitive to the value of the bending rigidity. In this regime the relaxation time is dependent on the permeability and surface tension, and we find

τ(A)BL ∼ R02 ⁄ (λpγ0)

(14)

The crossover between bending rigidity dominated and surface tension dominated relaxation occurs at a characteristic droplet radius Rc1 ) (k/γ0)1/2. Table 1 lists values for the compositions, viscosities, surface tension, and bending rigidity of a gelatin-dextran-water system studied by Scholten et al.2,3 The overall composition of this system prior to phase separation was close to the composition at the critical point. In Figure 1 we have used the values in Table 1 to calculate the longest relaxation time, at three different values of the permeability λp. The middle curve (]) represents a value for λp equal to 3.4 × 10-3 m3/(N s), a value determined by measuring the relaxation time for this system in cessation of flow experiments.2,3 The top and bottom curves are calculated to illustrate the effect of mass transfer on the relaxation time. We see that when the permeability increases from a value of 1.0 × 10-4 m3/(N s) to 5.0 × 10-2 m3/(N s), the relaxation time decreases by almost 3 orders of magnitude. In this regime mass transfer has a significant effect on the relaxation behavior, and neglecting this effect in the analysis of experimental data will lead to erroneous results.

13506 J. Phys. Chem. B, Vol. 112, No. 43, 2008

Sagis crossover length depends on the concentrations of the various components in the system, and we may have a different value for Rc2 for each component. For the system in Table 1, Rc2 equals 4.9 × 10-7 m for water and 4.5 × 10-5 m for both dextran and gelatin. From (15) and (16) we find two additional scaling regimes when δR0 ≈ R0. In that case the scaling for the viscosity-bending rigidity dominated relaxation reduces to

τ(A)BL ∼ R03∆Fj(A) ⁄ (Bk)

(17)

and the scaling in the viscosity-surface tension dominated regime reduces to

τ(A)BL ∼ R0∆Fj(A) ⁄ (Bγ0) Figure 3. Scaling of the relaxation time τ(A)BL with droplet radius R0, and deformation amplitude δR0, for the sample in Table 1. In regime A we find τ(A)BL ∼ R04/(λpk); in B we observe τ(A)BL ∼ R02/(λpγ0); C represents the regime where τ(A)BL ∼ R04∆Fj(A)/(δR0Bk); in D we find τ(A)BL ∼ R02∆Fj(A)/(δR0Bγ0); in E we observe τ(A)BL ∼ R03∆Fj(A)/(Bk); and j F(A)/(Bγ0). in F we find τ(A)BL ∼ R0∆

In Figure 1 we also clearly see the two scaling regimes: for small droplets the slope of the curve is equal to 4, and for larger droplets the slope is equal to 2. For the system in Table 1 the crossover between bending rigidity and surface tension dominated relaxation occurs at a characteristic droplet radius Rc1 equal to 2.03 × 10-6 m. In Figure 2 we compare the relaxation time calculated with (10) and the data in Table 1, with experimental data for the relaxation time determined in cessation of flow experiments.2,3 The experimental data are given by the open squares, and the solid curve represents eq 10 with a fitted value for λp equal to 5.5 × 10-3 m3/(N s). We see that there is good agreement between the experimental results and the model. For the experimental data the slope of the curve is equal to 1.8, very close to the scaling predicted by eq 14. This implies we are in the regime where the relaxation is dominated by the interfacial permeability and surface tension. The dashed line in Figure 2 is the scaling observed for impermeable interfaces, τ ∼ R0η/γ0, derived by Oldroyd.18 We clearly see that the scaling observed in the gelatin-dextran system deviates significantly from this result. The transition to the bending dominated regime could not be observed in these experiments. For this particular system this regime ends at droplet sizes of about a micron, with relaxation times below 0.001 s. The droplets were deformed using a Linkam shear cell (Linkam Scientific Instruments, type CCS 450), mounted on a Zeis Aksioskop 2 microscope with a Hitachi CCD color camera. The optical and time resolution of this setup was insufficient to measure droplet relaxation times shorter than 100 ms. To summarize, for small deformations, i.e., δR0 , λp∆Fj(A)/ B, we see two distinct scaling regimes, where relaxation is either permeability-bending rigidity dominated or permeability-surface tension dominated. For large deformations, i.e., δR0 . λp∆Fj(A)/ B, the φ(i) terms in (10), containing the viscosities of the bulk phases, dominate the permeability term. In this regime we also find two distinct scaling regimes. We find either viscosity-bending rigidity dominated relaxation, with

τ(A)BL ∼ R04∆Fj(A) ⁄ (δR0Bk)

(15)

For a system with very low permeability, like for example a viscous oil droplet dispersed in a water phase, eq 18 reduces to the familiar result τ ∼ R0η/γ0.18 The scaling behavior in eq 17 has recently been observed in the coarsening of bicontinuous regions during phase separation of gelatin-maltodextran-water systems.19 In these systems a transition from bending-rigidity driven hydrodynamic flow to surface tension driven hydrodynamic flow occurs at a characteristic length scale L equal to about 4 µm.20 This transition occurs at roughly the same characteristic length as in the system in Table 1. In Figure 3 all six scaling regimes which can be observed in phase-separated biopolymer solutions are plotted for the system in Table 1. When the surface tension, bending rigidity, permeability, viscosities, and compositions of a system are known, we can easily construct this type of parameter map. The vertical line is given by the value of Rc1 and the horizontal line by the value of Rc2. For Rc2 we have used the value for water (4.9 × 10-7 m). The curves bounding regimes E and F, where δR0 ≈ R0, are merely a rough estimate. These maps are very useful, since they allow us to predict which parameters will dominate the dynamic behavior of the system. For example, they illustrate how the scaling behavior can change during phase separation in the gelatin-maltodextran-water system studied by Lore´n et al.19 The scaling of the dynamics of that system was most likely at short times described by regime A, although with the microscopic technique used by the authors to examine the structural evolution of the system (confocal scanning laser microscopy), this regime could not be observed. As the characteristic length scales increased during phase separation, the dynamics of the system showed a transition to the scaling in regime E and finally to the scaling in regime F. So the phase separation process can be described as a path in the scaling plot, with time as a parameter, starting somewhere in the lower left corner of the plot and ending in the upper right corner. Hydrogel Beads. Hydrogel beads differ from phase-separated biopolymer systems in that the interior phase is a viscoelastic gel rather than a viscous fluid. Repeating the analysis from the previous section, but now with eq 7 as the constitutive equation for the extra stress tensor of phase 2, we find that the longest relaxation time for a hydrogel bead is given by (Appendix B; again assuming the effects of the spontaneous curvature are negligible)

τ(A)HL )

(16)

The crossover from permeability to viscosity dominated relaxation occurs at a characteristic length Rc2 ) λp∆Fj(A)/B. This

2R04

1

(1) (2) λp + φ(A) + φ*(A) 2γ0R02 + k

or viscosity-surface tension dominated relaxation, with

τ(A)BL ∼ R02∆Fj(A) ⁄ (δR0Bγ0)

(18)

(i) φ*(A) )

(i) Fj(A) δR0

∆Fj(A)M(i)(ω)

(19)

(20)

where M(i)(ω) ) ν(i)/(ω) + 2η(i)/(ω). Since both ν(i)/(ω) and η(i)/(ω) are complex numbers, the relaxation time τ(A)HL is

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complex. As we pointed out in the Introduction, this implies that hydrogel beads can have an oscillatory response to a deformation. Whether these oscillations are observable depends (2) (1) on the relative magnitude of φ*(A) with respect to λp and φ(A) . More precisely, an oscillatory response can be expected when (2) (2) (1) δR0 . λp∆Fj(A)M(2)/Fj(A) and M(2) , κ(1)Fj(A) /Fj(A). The longest relaxation time can then be separated in a real and imaginary ′ ′′ part: τ(A)HL ) τ(A)HL - iτ(A)HL . For the real and imaginary parts we find  τ(A)HL )

 τ(A)HL )

∆Fj(A)(ν(2)′(ω) + 2η(2)′(ω))

2R04

(2) Fj(A) δR0

2γ0R02 + k

∆Fj(A)(ν(2)′′(ω) + 2η(2)′′(ω))

2R04

(2) Fj(A) δR0

2γ0R02 + k

(21)

(22)

where we have used η(2)/(ω) ) η(2)′(ω) - iη(2)′′(ω).11 In this expression η(2)′(ω) is the viscous contribution to the complex shear viscosity, and η(2)′′(ω) is the elastic contribution to the complex shear viscosity. Similarly, we have also separated ν(2)/ 2 (2) ) η(2) b * - /3η * in a viscous and elastic contribution. The complex shear viscosity is related to the complex shear modulus by G(2)/(ω) ) iωη(2)/(ω). The complex shear modulus can be determined using classical rheological experiments11,12 or microrheological experiments. Similarly, we find for the complex (2) 21 (2)/(ω) is bulk viscosity iωη(2) b *(ω) ) K *(ω) - K0, where K the complex bulk modulus of the gel bead, and 1/K0 is its adiabatic compressibility.21 For these parameters only limited data is available in the literature. When the relaxation time is given by (21) and (22), we can observe two scaling regimes. For small beads, i.e., R0 , (k/ ′ γ0)1/2, we find that τ(A)HL depends on the viscosities η′ and η′b and the bending rigidity k. For R0 . (k/γ0)1/2 the real part of the relaxation time is determined by the same viscosities and the surface tension. (2) (2) (1) When φ*(A) , λp and φ*(A) , φ(A) , we will not observe an oscillatory response to a deformation, and we will observe the same six scaling regimes we found in the previous section for phase-separated biopolymer systems. Note that eq 19 can easily be extended to a system were the continuous phase 1 is a semidilute polymer solution. In that case the continuous phase would also exhibit viscoelastic (1) rheological behavior. For such a system we simply replace φ(A) (1) (1) by φ*(A) , where φ*(A) is given by eq 20. Conclusions We have seen that the longest relaxation time of a perturbation of the interfaces in phase-separated biopolymer solutions and hydrogel beads may depend on surface tension, bending rigidity of the interface, interfacial permeability, viscosities of the bulk phase, and shear and bulk modulus of the bulk phases. Depending on the values of these parameters, we can observe a wide range of scaling behaviors of the relaxation time, with the size of the droplet, and its deformation. In phase-separated biopolymer solutions and hydrogel beads we can observe up to six scaling regimes, in which τ(A) ∼ R0n, and n varies from 1 to 4. We have shown how to construct parameter maps for these systems that allow us predict which parameters will dominate the relaxation behavior of these systems. The decay of fluctuations of the interfaces in water-in-water emulsions may be multiexponential, and in the case of hydrogel beads, we may also have a damped oscillatory response. These oscillations can greatly enhance mass transfer from these

systems, which will influence their performance when used for encapsulation and controlled release purposes. The results presented here provide new insight into the complex dynamics of water-in-water emulsions. They also suggest alternative methods to characterize the interfaces in water-in-water emulsions. By deforming droplets in either an imposed flow or another appropriate external field (for example, using electromagnetic fields or ultrasound),22 the relaxation time can be measured as a function of droplet radius, using for example microscopy combined with high-speed cameras,2,3,22 small-anglelightscattering,23 orlineardichroismmeasurements.23,24 By plotting the relaxation time versus droplet radius, the appropriate scaling regime can be identified, and the essential interfacial parameters can be obtained by fitting the experimental results with the expressions derived here. Appendix A. Relaxation Time for Phase-Separated Biopolymer Mixtures To derive an expression for the relaxation time for the perturbation of an interface in a phase-separated biopolymer mixture, we introduce the Fourier transformation

g(q, ω) )

∫-∞∞ eiωt dt∫ g(r, t)eiq · r dr

(23)

After applying this transformation to (2) and eliminating the velocity fields using the transform of (4) and (5), we find (1) (2) iω∆Fj(A)R(q, ω) + λp∆Fj(A)p0(q, ω) - ∆Fj(A)[λp + φ(A) + φ(A) ]× (1) (2) P (q, ω) + (P(1)(q, ω) - P(2)(q, ω)) - ∆Fj(A)φ(A) (2) (1) (1) (1) h(1) P (q, ω) - Fj(A) Orj fj (q, ω) + ∆Fj(A)φ(A) (2) (2) h(2) Orj fj (q, ω) ) 0 (24) Fj(A) (i) where φ(A) is given by

(i) φ(A) )

(i) Fj(A)

(25)

∆Fj(A)κ(i)qr

The function p0(q, ω) equals 2γR0-1δ(ω)δ(q), qr is the r4 (i) component of q, and κ(i) ) ν(i) + 2η(i) ) η(i) b + /3η . The tensor O(i) is given by

Oij(i) )

(

1 ν(i) + η(i) qiqj δij η q κ(i) q2 (i) 2

)

(26)

For phase-separated biopolymer mixtures we neglect contributions from the surface extra stress tensor in (3). We will also neglect the contribution from the spontaneous curvature C0. The transform of this equation (3) is then given by

1 -q|2 γ0 + kq|2 R(q, ω) + P(1)(q, ω) - P(2)(q, ω) + 2

(

)

(2) (1) (q, ω) - σrr (q, ω) ) 0 (27) σrr

where q| is the magnitude of the component of the q vector, parallel to the interface. For a weakly deformed interface q|2 ) qθ2 + qφ2, where qθ and qφ are the θ- and φ-components of the q vector. Equation 27 is used to eliminate the pressure difference (P(1) - P(2)) in (24). We find (1) (iω - τ(A)-1)R(q, ω) + λp p0(q, ω) - Θ(σrr (q, ω) (2) (1) (2) (2) (1) (q, ω)) - φ(A) P (q, ω) + φ(A) P (q, ω) σrr (1) (1) h(1) (2) (2) h(2) Ozj fj (q, ω) + z(A) Ozj fj (q, ω) ) 0 (28) z(A) (i) (i) where z(A) ) Fj(A) /∆Fj(A), and τ(A) is the relaxation time of the perturbations of the interface, given by

13508 J. Phys. Chem. B, Vol. 112, No. 43, 2008

Sagis

1 1 ) Θq|2 γ0 + kq|2 τ(A) 2

(29)

(1) (2) Θ ) [λp + φ(A) + φ(A) ]

(30)

(

)

and

Appendix B. Relaxation Time for Hydrogel Beads For hydrogel beads the inner phase, here referred to as phase 2, is assumed to be a linear viscoelastic material. We can repeat the analysis of the previous section to find that eq 28 now takes the form (1) (iω - [τ(A)H]-1)R(q, ω) + λpp0(q, ω) - Θ*(σrr (q, ω) (2) (1) (2) (2) (1) (q, ω)) - φ(A) P (q, ω) + φ*(A) P (q, ω) σrr (1) (1) h(1) (2) (2) h(2) Ozj fj (q, ω) + z(A) Wzj fj (q, ω) ) 0 (31) z(A) (1) where φ(A) is still given by (25)

(i) φ*(A) )

and

Wij(i) ) Here

(i) Fj(A)

(32)

∆Fj(A)(ν(i) * (ω) + 2η(i) * (ω))qr

(

)

1 ν(i) * (ω) + η(i) * (ω) qiqj δ (33) ij η(i) * (ω)q2 ν(i) * (ω) + 2η(i) * (ω) q2

η(i)*(ω)

is the complex shear viscosity, defined by11

∫-∞∞ G(t)eiωt dt

(34)

∫-∞∞ (K(t) - 32 G(t))eiωt dt

(35)

η(i) * (ω) ) and ν(i)*(ω) equals

ν(i) * (ω) )

The relaxation time for the perturbations of the interface is now given by

1 1 ) Θ*q|2 γ0 + kq|2 τ(A)H 2

(36)

(1) (2) Θ* ) [λp + φ(A) + φ*(A) ]

(37)

(

)

From (36) and (37) we find that the longest relaxation time for a spherical droplet with radius R0 is given by eq 19.

References and Notes (1) Sagis, L. M. C. J. Controlled Release 2008, 131, 5. (2) Scholten, E.; Sprakel, J.; Sagis, L. M. C.; van der Linden, E. Biomacomolecules 2006, 7, 339. (3) Scholten, E.; Sagis, L. M. C.; van der Linden, E. J. Phys. Chem. B 2006, 110, 3250. (4) Scholten, E.; Sagis, L. M. C.; van der Linden, E. Biomacomolecules 2006, 7, 2224. (5) Gordon, V. D.; Chen, X.; Hutchinson, J. W.; Bausch, A. R.; Marquez, M.; weitz, D. A. J. Am. Chem. Soc. 2004, 126, 14117. (6) Yonese, M.; Sugie, S.; Nagata, T.; Kiyohara, K. Effects of dynamic process of hydrogel on solute release. Chem. Lett. 1995, 917. (7) Slattery, J. C.; Sagis, L. M. C.; Oh, E.-S. Interfacial Transport Phenomena, 2nd ed.; Springer: New York, 2007. (8) Helfrich, W. Z. Naturforsch. C 1973, 28, 693. (9) Sagis, L. M. C. Phys. ReV. Lett. 2007, 98, 066105. (10) Sagis, L. M. C. J. Phys. Chem. C 2007, 111, 3139. (11) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids, 2nd ed.; Wiley-Interscience: New York, 1987; Vol. 1. (12) Macosko, C. W. Rheology: Principles, Measurements, and Applications; Wiley-VCH: New York, 1994. (13) Prost, J.; Manneville, J.-B.; Bruinsma, R. Eur. Phys. J. B 1998, 1, 465. (14) Huang, J. S.; Milner, S. T.; Farago, B.; Richter, D. Phys. ReV. Lett. 1987, 59, 2600. (15) Farago, B.; Richter, D.; Huang, J. S.; Safran, S. A.; Milner, S. T. Phys. ReV. Lett. 1990, 65, 3348. (16) Gang, H.; Krall, A. H.; Weitz, D. A. Phys. ReV. Lett. 1994, 73, 3435. (17) Gang, H.; Krall, A. H.; Weitz, D. A. Phys. ReV. E 1995, 52, 6289. (18) Oldroyd, J. G. Proc. R. Soc. London 1953, 218, 122. (19) Lore´n, N.; Altska¨r, A.; Hermansson, A.-M. Macromolecules 2001, 34, 8117. (20) Scholten, E.; Sagis, L. M. C.; van der Linden, E. Macromolecules 2005, 38, 3515. (21) Boon, J. P.; Yip, S. Molecular Hydrodynamics; Dover Publications: New York, 1980. (22) Bouakaz, A.; Versluis, M.; de Jong, N. Ultrasound Med. Biol. 2005, 31, 391. (23) vermant, J.; van Puyvelde, P.; Moldenaars, P.; Mewis, J. Langmuir 1998, 14, 1612. (24) van Puyvelde, P.; Moldenaars, P.; Mewis, J. Phys. Chem. Chem. Phys. 1999, 1, 2505.

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