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Gradient Sensing in Reactive, Ternary Membranes Olga Kuksenok and Anna C. Balazs* Chemical Engineering Department, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261 ReceiVed September 14, 2007. In Final Form: NoVember 26, 2007 Using computer simulations, we investigate the behavior of reactive ternary ABC membranes that are subjected to an external, spatially nonuniform stimulus, which controls the rate of interconversion between the A and B components. We assume that A and B have different spontaneous curvatures. Furthermore, the C component is taken to be nonreactive and incompatible with both A and B. We find that a gradient in the applied stimulus causes the dynamic reconstruction of the membrane, with a preferential reorientation of the reactive AB domains along the gradient. In addition, the external gradient effectively controls the transport of the nonreactive C component within the membrane. The latter effect could potentially be exploited for cleaning the membrane of the nonreactive C “impurities” or for the targeted delivery of the C component to specific locations.
Introduction Due to their inherent physical diversity and chemical reactivity, biological membranes can perform a variety of functions, such as sensing, signaling, molecular recognition, and transport. Theoretical studies of model systems can enhance our understanding of various dynamical processes occurring within biological membranes and can provide design rules for creating synthetic membranes that respond to applied external stimuli by, for example, undergoing dynamic restructuring or self-healing.1 Despite significant progress that has been made in understanding the behavior of both biological and synthetic membranes,2,3 the nonequilibrium dynamics of reactive, multicomponent membranes has received far less attention. One observation that has emerged from the existing studies4-9 in this area is that different active inclusions (such as proteins) can affect the shape of the membrane. Such inclusions (i.e., proteins) can be activated by an external light source,5 and thus, the external stimuli can be used to dynamically control the shape of the membrane. Furthermore, experiments show that the magnitude and the sign of the membrane’s curvature could be controlled by photoinduced chemical reactions in aqueous solutions.10 Here, we focus on designing model synthetic membranes that respond to an applied, spatially nonuniform stimulus by performing two functions: (a) dynamic reconstruction that leads to a specific architecture, which is defined by the value of the gradient in the stimulus, and (b) transport of nonreactive inclusions in a direction and with a speed defined by this gradient. The latter function can be of particular importance not only for the controlled transport of nonreactive species to specific locations but also for the self-cleaning of the membrane by means of expelling all impurities to a designated region. Our study is based on the phenomenological model recently introduced by Reigada et al.11,12 that captures the dynamics of (1) Parikh, A. N.; Groves, J. T. MRS Bull. 2006, 31, 507. (2) Lipowsky, R.; Sackmann, E. The Structure and Dynamics of Membranes; Elsevier: Amsterdam, 1995. (3) Mueller, M.; Katsov, K.; Schick, M. Phys. Rep. 2006, 434, 113. (4) Ramaswamy, S.; Toner, J.; Prost, J. Phys. ReV. Lett. 2000, 84, 3494. (5) Manneville, J. B.; Bassereau, P.; Levy, D.; Prost, J. Phys. ReV. Lett. 1999, 82, 4356. (6) Manneville, J. B.; Bassereau, P.; Ramaswamy, S.; Prost, J. Phys. ReV. E 2001, 64, 021908. (7) Chen, H. Y. Phys. ReV. Lett. 2004, 92, 168101. (8) Chen, C. H.; Chen, H. Y. Phys. ReV. E 2006, 74, 051917. (9) Gov, N. Phys. ReV. Lett. 2004, 93, 268104. (10) Petrov, P. G.; Lee, J. B.; Dobereiner, H. G. Europhys. Lett. 1999, 48, 435. (11) Reigada, R.; Buceta, J.; Lindenberg, K. Phys. ReV. E 2005, 71, 051906.
a reactive bilayer in which two differently shaped components, A and B, undergo the following interconversion reaction in the presence of an external stimulus: Γ+
A S B Γ-
(1)
Here, Γ+ and Γ- are the respective reaction rate coefficients for the forward and reverse reactions. The researchers showed11,12 that in the case where the A and B components phase separate, the bilayer forms stationary patterns of spatially different composition and curvature; the characteristic length scale of these patterns is defined by the above reaction rate coefficients. In this system, the pattern formation occurs due to a competition between the reaction and phase separation. We note that both hexagonal and lamellar-like patterns can be observed in a variety of mixtures that encompass competing interactions. For example, such ordered structures appear in mixtures of oppositely charged molecules due to competition between phase separation and electrostatic interactions,13 and in double bilipid membranes14 due to the phase separation and the interaction between the two membranes. We recently extended the approach of Reigada et al.11,12 by explicitly including the effects of the membrane’s surface tension on the phase behavior of the system.15 We also considered threecomponent ABC membranes and showed that the presence of the nonreactive C component strongly affects the composition and topology of the membrane.15 Through these studies, we isolated novel dynamical patterns that arise due to the interplay between the A-B interconversion reaction and the phase separation among all three components. In the above studies, it was assumed that the external stimulus was applied uniformly over the entire sample. For example, the membrane was irradiated evenly by an external light source. Consequently, the value of the reaction rate coefficients did not exhibit a spatial dependence. While there have been no prior investigations of such reactive membranes in nonuniform external fields, our previous studies of flat, multicomponent films revealed that the application of a spatially varying stimulus (e.g., light source) could provide significant control over the structure formation in the system.16-18 In this paper, we use our (12) Reigada, R.; Buceta, J.; Lindenberg, K. Phys. ReV. E 2005, 72, 051921. (13) Loverde, S. M.; Solis, F. J.; Olvera de la Cruz, M. Phys. ReV. Lett. 2007, 98, 237802. (14) Sample, C.; Golovin, A. A. Phys. ReV. E 2007, 76, 031925. (15) Kuksenok, O.; Balazs, A. C. Phys. ReV. E 2007, 75, 051906.
10.1021/la7028615 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/26/2008
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computational model to focus on the response of ABC reactive membranes to an external stimulus that produces a gradient in the value of the rate coefficients along the length of the sample. In particular, we determine the effects of this external gradient on the dynamic reconstruction of the membrane and the transport of the nonreactive C component within the membrane.
The Model To carry out these studies, we consider a flexible membrane composed of a ternary ABC mixture, where all the components are mutually immiscible. The height of the system, h ≡ h(x, y), is calculated with respect to a flat, undistorted membrane. In response to an external stimulus, the A and B components undergo the reversible chemical reaction in eq 1. Here, we assume that Γ+ ) Γ-. We further assume that the A and B domains exhibit different spontaneous curvatures.11,12 Such membranes could represent a polymeric bilayer, and A might be a group of lipids, biomolecules, or isomers of an amphiphilic azobenzene derivative that exhibits a particular curvature, while the B group displays a different curvature.11,12 The C component might be any nonreactive group, or even an impurity, that is immiscible with both the A and B components.15 The composition of the membrane can be described in terms of two order parameters: φ is difference between the number densities of the A and B components, and ψ is the number density of the C component. The form of the free energy functional F is taken to be15
F(φ, ψ, h) )
[
∫ dr f0(φ, ψ) + σ2(∇h)2 +
γφ (∇φ)2 + 2
]
γψ κ (∇ψ)2 + (∇2h - φH0)2 (2) 2 2
The first term in eq 2 is the local free energy density and is given by f0(φ, ψ) ) -a20φ2 + a40φ4 + a02ψ2 - a03ψ3 + a04ψ4 + a22φ2ψ2. The coefficients aij in this expression were chosen to ensure that the entire system undergoes phase separation.19,20 In particular, we chose the coefficients aij to have the following values: a40 ) 0.5a20, a02 ) 3a20, a03 ) 8a20, a04 ) 4.5a20, a22 ) 3a20, and a20 ) 0.5. As a consequence, f0(φ, ψ) has equal minima at φ ) (1, ψ ) 0 (pure A(B) phase) and φ ) 0, ψ ) 1 (pure C phase).20 The second term in eq 2 (σ-term) describes the effects of the lateral surface tension of the membrane. The third term (γφterm) accounts for the interfacial tension between the A and B components, and similarly, the fourth term (γψ-term) accounts for the interfacial tension between C and the A(B) components. The last term in eq 2 specifies the elastic energy of the film due to its rigidity, where κ is the bending modulus.3 In this last term, we assumed a linear dependence of the local equilibrium curvature on the composition,11,12 that is, Heq(φ) ≡ φH0. With respect to the ternary mixture, this choice for the local curvature ascribes a preferentially flat topology for the C component.15 (16) Travasso, R. D. M.; Kuksenok, O.; Balazs, A. C. Langmuir 2006, 22, 2620. (17) Travasso, R. D. M.; Kuksenok, O.; Balazs, A. C. Langmuir 2005, 21, 10912. (18) Kuksenok, O.; Travasso, R. D. M.; Balazs, A. C. Phys. ReV. E 2006, 74, 01152. (19) Good, K.; Kuksenok, O.; Buxton, G. A.; Ginzburg, V. V.; Balazs, A. C. J. Chem. Phys. 2004, 121, 6052. (20) Travasso, R. D. M.; Buxton, G. A.; Kuksenok, O.; Good, K.; Balazs, A. C. J. Chem. Phys. 2005, 122, 194906.
The evolution of the film is described by the following set of equations15,20-22
[
∂f0(φ, ψ) ∂φ ) Mφ∇2 - γφ∇2φ + ∂t ∂φ
]
κH02φ - κH0∇2h - Γφ (3)
[
∂f0(φ, ψ) ∂ψ ) Mψ∇2 - γψ∇2ψ ∂t ∂ψ
]
∂h ) Mh∇2[σh + κφH0 - κ∇2h] ∂t
(4) (5)
where Mi is the mobility of the ith order parameter and Γ ≡ Γ+/2 ) Γ-/2. If the C component is absent and the surface tension in the membrane is zero (σ ) 0), the above model reduces to the one proposed in ref 11. Furthermore, if there is no C component and no spontaneous curvature associated with the A(B) components, and the membrane remains flat at all times (h ≡ 0), the above model reduces to a well-known model for block copolymers23,24 or reactive polymer blends.25,26 For the case of flat, binary films (h ≡ 0), it has been shown23,26 that eq 3 can be rewritten as ∂φ/∂t ) Mφ∇2 δF ˜ (φ, h)/δφ, where
F(φ, h) ) F(φ, h) +
Γ Mφ
∫∫ dr dr′ G(r - r′) φ(r) φ(r′)
(6)
and the Green’s function G(r - r′) can be found from Poisson’s equation ∇2 G(r - r′) ) -δ(r - r′) with the appropriate boundary conditions. Below, we will use eq 6 to estimate the height of the membrane at a given value of Γ and, consequently, estimate the response of the membrane to the imposed external gradient in Γ. We note that the second term on the right-hand side (rhs) of eq 6 does not depend on h; that is, we can readily apply the expression derived for the case of the flat, binary reactive film to the current case of a membrane with variable height (see below). To obtain the results presented below, we numerically integrated eqs 3-5.15 We assumed that the membrane height is fixed at all the boundaries at h ) 0; we also assumed that all the boundaries are neutral and that there is no flux of the components through the boundaries. Unless specified otherwise, we used the following values for the dimensionless parameters: for the binary mixture, Mφ ) 1, Mh ) 1, κ ) 0.5, H0 ) 0.2, γφ ) 111,12, and σ ) 0.0515 and, for the C component, Mψ ) 1/3 and γψ ) 3.15,20
Results and Discussion We consider the evolution of the three-component membranes subjected to the following gradient of reaction rates along the (21) In the more general case, the externally controlled reaction can directly affect not only the local composition but also the local shape of the membrane,12 and such an effect would lead to the addition of the appropriate reactive term into the right-hand side of eq 5; in the present work, we neglect such effects. (22) Here, we neglect hydrodynamic interactions in the system; this is valid for a membrane immersed in a highly viscous medium. For a system in which hydrodynamic effects cannot be neglected, an overdamped relaxational dynamics of the membrane height would no longer be valid (e.g., see discussion in ref 12 and references therein). In addition, the advection of both order parameters due to the hydrodynamic effects should be incorporated. It was previously shown (see, e.g., Huo, Y.; Jiang, X.; Zhang, H.; Yang, Y. J. Chem. Phys. 2003, 118, 9830) that hydrodynamic interactions could significantly affect the pattern formation in binary, phase-separating, flat films undergoing the reaction in eq 1. Therefore, it is reasonable to expect that the patterns formed in the ternary reactive membrane would also be affected by the hydrodynamic interactions. (23) Liu, F.; Goldenfeld, N. Phys. ReV. A 1989, 39, 4805. (24) Bahiana, M.; Oono, Y. Phys. ReV. A 1990, 41, 6763. (25) Glotzer, S. C.; Dimarzio, E. A.; Muthukumar, M. Phys. ReV. Lett. 1995, 74, 2034. (26) Christensen, J. J.; Elder, K.; Fogedby, H. C. Phys. ReV. E 1996, 54, R2212.
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Figure 1. Distribution of (a) φ(x, y) and (b) h(x, y). The simulation box size is Lx × Ly ) 190 × 120. In (a-e), Γmin ) 0.08, Γmax ) 0.18, and the simulation times are (a and b) t ) 104, (c) t ) 2 × 105, (d) t ) 106, and (e) t ) 3 × 106. In (f and g), Γmin ) 0.12, Γmax ) 0.18. The radius of the circular C domain at t ) 0 is Rc ) 44 in (a-g) and Rc ) 38 in (h). H0 ) 0.2 in (a-f) and (h), and H0 ) 0.5 in (g).
x-direction: Γ(x) ) Γmin + (Γmax - Γmin )x/Lx (see Figure 1a). In all of the simulation results presented below, we set the reaction rates to have a minimum value, Γmin, at the left side and a maximum value, Γmax, at the right side of the membrane. First, we focus on the scenario where a single domain of the nonreactive C component is initially located on the left side of the membrane (lower values of Γ). We then consider the scenario where all three components are initially intermixed and undergo phase separation in the presence of the external gradient. We show that the C component migrates along the gradient in Γ and, simultaneously, the AB domains reorient along this gradient. We also provide an analytical estimate for the average height of the AB membrane as a function of the bending elasticity and surface tension of the membrane, and the spontaneous curvature of the components. We then compare these predictions with the results of our simulations. Figure 1 depicts the time evolution of a membrane with a single C domain in the presence of the spatially varying Γ. Initially, the membrane was flat; that is, h(r, t)|t)0 ) 0 and the A and B components were intermixed, with a small random fluctuation of φ(r, t) around its average value, 〈φ(r, t)〉 ) 0. The C component was initially placed as a single circular domain r ∈ RC; within this domain, we set the density of the C component to ψ(r, t)|r∈RC ) 1. The spatial distributions of the order parameter φ(r) and of the height of the membrane h(r) at a relatively early time are shown in Figure 1a and b, respectively; the C component occupies a large flat domain with φ(r) ) 0. Figure 1c-e illustrates that this membrane “responds” to the application of the gradient in Γ by the following: transporting the C domain along the gradient, reshaping this domain from circular to rectangular, and preferentially reorienting the AB domains along the gradient.27 We note that, at early times (see Figure 1a and b), all the AB domains are preferentially oriented
perpendicular to all the boundaries (recall that we have neutral boundary conditions at all the boundaries). In addition, the AB domains are preferentially oriented perpendicular to the edges of the C domain; this is due to the fact that the C interface effectively acts as a neutral boundary with respect to the order parameter φ (see refs 16-18). At later times, as the C domain moves along the gradient, the preferential perpendicular orientation at the left boundary propagates throughout the sample (Figure 1c-e). We also note the appearance of some defects in the reoriented structure; these arise because the membrane is attempting to accommodate a decrease in the characteristic width of the domains, w, along the gradient (see eq 7a) within the fixed nonperiodic boundaries. We note that the diffusion of the C component toward the region of the higher reaction rates occurs also within a flat membrane (e.g., in a tensionless membrane in which the spontaneous curvatures for both the A and B components are zero). In such a case, however, there is no change in the topology of the membrane (it remains flat at all times) and the underlying physics that controls the migration of C to higher Γ is the same as that described in our studies of pattern formation in ABC thin films and bulk blends.16-18 In the latter systems, two regions with distinct values of Γ were present within the sample and the C component was observed to diffuse to the region with the higher Γ. The localization of C to the higher Γ region minimized the total free energy of the system.16-18 In particular, at higher Γ, the characteristic widths of the A and B domains (w ∼ 1/k) are narrower (eq 7a) and more intermixed, resulting in a higher (27) Additional simulations show that the reorientation of the AB domains along the external gradient is a very robust process that occurs in binary AB reactive films as well; however, in the latter case, the reorientation is observed to be slower because of the absence of the additional moving neutral interface (C domain).
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Figure 2. Evolution of h(x, y) for Γmin ) 0.12, Γmax ) 0.18 and radius of the circular C domain (at t ) 0) Rc ) 55. The simulation box size is Lx × Ly ) 190 × 120. The simulation times are (a) t ) 104, (b) t ) 5 × 104, (c) t ) 1.5 × 105, (d) t ) 106, (e) t ) 2 × 106, and (f) t ) 4 × 106.
interfacial energy in this region. The migration of the C component to the higher Γ regions displaces the A and B components, driving them to the lower Γ area, where the A and B domains are larger and less intermixed and, thus, exhibit a lower interfacial energy. It is in this manner that the migration of the C component lowers the total free energy of the system. It can be shown that, in the case of the reactive ternary membrane, the location of the C component within the region of higher Γ also minimizes the total free energy of the system. In addition, the presence of the imposed gradient permits smooth “sliding” of the C component along the gradient, with a velocity that is defined by the difference in Γ along the x-direction, as we illustrate further below. The images in Figure 1f-h illustrate different steady-state scenarios for membranes subjected to smaller gradients in Γ. While the properties of the membranes themselves in Figure 1f and 1a-e are the same, the presence of a smaller gradient in Γ leads to more ordered and aligned domains28 than in the previous case; this is due to the fact that the decrease in the characteristic width of the domains along the gradient is smaller. Figure 1g shows the steady-state structure of a membrane with H0 ) 0.5, so that the spontaneous curvature of the A and B components is higher than in the previous cases. Here, the defects in the ordering of the AB domains are more pronounced close to the left boundary (i.e., higher Γ). And finally, Figure 1h corresponds to the case of a smaller C domain. In this scenario, the C domain no longer forms a stripe at higher values of Γ, but instead it occupies one of the right-hand corners. We note that sufficiently large C domains (such that the diameter of the domain is very close to the width of the membrane in the y-direction) can exhibit a shape change even at relatively early times, as seen in Figure 2, which shows a large circular domain transforming into a stripe. Recall that the boundaries of the membrane are flat and neutral, so that the interfaces between the C and A/B regions align perpendicular to the edge of the membrane and, consequently, the reshaping of a sufficiently large C domain into a stripe that extends across the sample reduces the interfacial energy of the system. After this C stripe is formed, it continues to slide along the gradient, leaving the ordered AB domains on the left (Figure 2). If we increase the width of the membrane while keeping all the other parameters (including the (28) With different initial conditions, alignment can be slightly less perfect due to some defects that appear and “freeze” close to the boundaries.
size of the C domain) fixed, the transformation of the C domain from a circular shape to a stripe (as in Figure 2) would no longer occur and the C domain would keep its circular shape while diffusing along the membrane (i.e., the system would behave as in the example in Figure 1). To quantitatively characterize the dynamics of such membranes, we calculate the position of the center of mass of the C domain along the x-axis, R, and plot its value in Figure 3. Figure 3a shows that the time needed for the C domain to reach the right boundary of the membrane depends critically on the gradient in Γ. Even a small decrease in the value of Γmin (while keeping Γmax fixed) causes a rather dramatic speed up of the process and illustrates the high sensitivity of the membrane to the external gradient. For example, a 17% decrease in the value of Γmin causes a roughly 2-fold increase in the speed at which the C domain saturates the right side of the membrane (compare the curves with Γmin ) 0.12 and Γmin ) 0.1 in Figure 3a). In other words, the use of external gradients allows us to control both the direction and speed of motion of the nonreactive components within the membrane. It is also worth noting that the error bars in Figure 3a are smaller for the case of larger gradients and are larger for the smaller gradient. This reflects the fact that, for the larger gradients in Γ, the driving force is sufficiently strong to cause a rather smooth migration of the C component along the gradient. For a relatively small gradient (see, e.g., the curve corresponding to the highest Γmin in Figure 3a), this driving force is smaller and the process depends more on the evolution of the distribution of the AB domains, which, in turn, depends on the initial fluctuations. In the latter case, the process can be significantly slower if the C domain shifts close to the boundary (y ) 0 or y ) Ly) and diffuses from left to right along this boundary instead of diffusing along the center of the membrane. Figure 3b shows that, for fixed Γmin and Γmax, the evolution of the center of mass of the C domain remains approximately the same for different values of the bending elasticity of the membrane, surface tension, and spontaneous curvature of the A and B components (even though the topology of the AB domains can be distinctly different). In the next series of simulations, we examined the evolution of the ternary membrane in an external gradient for the cases where all three components within the membrane are initially intermixed. An example of the evolution of such a system is
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Figure 3. (a) Evolution of the position of the center mass of the C domain along the x-direction, R, for the same membrane parameters but in different gradients of Γ. Each curve shows the average of three independent runs. Here, Γmax ) 0.18 and Γmin ) 0.08, 0.10, and 0.12 from left to right as marked on the legend. (b) Evolution of the position the center mass of the C domain along the x-direction, R, in the fixed gradient of Γ with Γmin ) 0.12 and Γmax ) 0.18 but for different membrane parameters as marked on the legend. Each curve shows the average of three independent runs. All other parameters are the same as in Figure 1a-f.
shown in Figure 4. As time increases, all three components undergo phase separation, which leads to an increase in the average size of the C component. The characteristic wavenumber for the AB pattern, however, is defined by the competition between the reaction and phase separation, and increases with an increase in Γ along the length of the membrane (see Figure 4 and estimates below). The diffusion of the C component along the gradient occurs simultaneously with the phase separation and reaction processes. In addition, similar to the scenario with the single C domain, the AB domains reorient along the gradient (see Figure 4). To predict the response of the AB membrane to the external gradient, we must estimate the average height of the membrane along the gradient in Γ. To do so, we first provide analytical calculations of the characteristic height of the AB membrane at a fixed value of Γ and then compare our analytical estimates with the simulation data for the membrane in the external gradient. We note that the behavior of binary phase-separating systems that undergo the reaction in eq 1 is well understood and analytical expressions for the characteristic wavelength and bulk order parameter as functions of the reaction rate coefficients have been derived for the case of the flat films;26 however, to date, there have been no derivations for the characteristic height of a binary membrane in the case where two interconverting components
Kuksenok and Balazs
Figure 4. Evolution of h(x, y) for Γmin ) 0.11, Γmax ) 0.2; the simulation box size is Lx × Ly ) 360 × 120, and 〈ψ〉 ) 0.27. The simulation times are t ) 104, t ) 3 × 104, t ) 3 × 105, t ) 1.1 × 106, and t ) 3.6 × 106 in (a-e), respectively. All other parameters are the same as in Figure 1a-f.
have nonzero curvatures. Here, we use the expression for the total free energy, eq 6, to estimate the height of the membrane, h, as a function of the membrane parameters for a given value of Γ. It was shown26 that, for the flat, binary films (h ≡ 0), one can use a single mode approximation for the order parameter and minimize the total free energy with respect to k and φ0 to obtain the characteristic wave number, k, (which is inversely proportional to the characteristic width of the domain) and the bulk value of the order parameter, φ0, as the following: 4
k ) xΓ/γφ φ0 )
x x 2a20 3a40
1-
(7a) 2 γ Γ a20x φ
(7b)
Even though these expressions were derived for the flat films, we carried out additional simulations of the binary membranes for different values of the bending elastic modulus and equilibrium curvatures and found that the above expressions for k and φ0 remain as valid approximations for the reactive membranes. Our results agree with the earlier studies of reactive membranes that showed that the equilibrium value of the domain size is the same for the flat (κ ) 0, H0 ) 0) and elastic (κ * 0, H0 * 0), tensionless (σ ) 0) membranes.11 In the current work, we propose to use a single mode 1D approximation for both the order parameter, φ ) φ0 cos kx, and the height of the membrane, h ) -h0 cos kx, with the values of
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and, thus, the fit between theory and simulation appears to be quite good along the whole length of the membrane. The sharp decrease in the height of the membrane (and in φ0 in the inset) at high Γ corresponds to the presence of the C stripe at the right boundary (as in Figure 4e). The inset to Figure 5 shows that the simulation data for the bulk value of the order parameter, φ0, corresponding to runs with different values of κ, H0, and σ merge onto a single curve given by the analytical estimates for φ0 (eq 7b). Thus, the effective height and the effective intermixing of the binary portion within the membrane subjected to the gradient in Γ can be estimated using the above analytical expressions for h0 and φ0. In other words, eqs 7b and 8 allow us to predict the response of the AB portion of the membrane to the imposed external gradient. Figure 5. Distribution of the membrane height, h0, along the applied gradient in Γ. Here, Γmin ) 0.11, Γmax ) 0.2, the simulation box size is Lx × Ly ) 360 × 120, and 〈ψ〉 ) 0.27. Each of the curves represents an average of three independent runs. The dashed lines represent analytical estimates (eq 8) for each set of parameters, that is, for H0 ) 0.5, κ ) 0.5; H0 ) 0.2, κ ) 0.5; and H0 ) 0.2, κ ) 0.2 from top to bottom, respectively, as marked on the figure. The inset shows the distribution of φ0 along the applied gradient in Γ for the same runs. The dashed line represents analytical estimates (eq 7). At the left boundary, we show the results from Γ(x) ) 0.12; that is, we cut off the boundary region (from Γ ) 0.11 to 0.12). As noted in the text, the sharp decrease in the height of the membrane (and in φ0 in the inset) at high Γ corresponds to the presence of the C stripe at the right boundary (as in Figure 4e).
k and φ0 taken as defined above for the flat films. This approximation is valid for sufficiently high values of Γ. Substituting the above approximations for φ and h into the total free energy (eq 6) and minimizing this free energy with respect to h0 then gives
x
a20 - xγφΓ 2 3a40 (κ Γ/γ + σ)
x
h0 ) H0κ
x
(8)
φ
To test these analytical predictions, in Figure 5, we plot the effective membrane height along the gradient in Γ for the different membrane parameters calculated from the simulation data (solid lines), as well as the estimates given by eq 8 for the same parameters (dashed lines). Here, for each value of x, we calculated 〈h0〉 ) π/2Ly ∫ |h(x, y)| dy along the x-direction using the simulation data for h(x, y) at late times (i.e., when all of the C domains have already migrated to the high Γ region). The horizontal axis in Figure 5 gives the values of Γ(x) corresponding to the position along the membrane to simplify comparison with the analytical expression for h0. The height of the membrane decreases with an increase in Γ within the binary domain in agreement with eq 8; as anticipated, these estimates are more accurate for the higher values of Γ. The latter observation is particularly evident in the blue curve in Figure 5. We note that the above analytical estimates involved no adjustable parameters
Conclusions In summary, we showed that application of an external gradient causes the dynamic reconstruction of a binary membrane, with the preferential reorientation of the AB reactive domains along the gradient. We also provided analytical estimates that allowed us to predict the topology that the reactive membrane assumes in response to the application of the external gradient. Furthermore, we found that the external gradient effectively controls the transport of the nonreactive C component within the membrane. We showed that the nonreactive components migrate along the gradient in Γ toward higher values of Γ and the speed of this migration is defined by the actual value of the gradient (the migration is faster for larger gradients). In the other words, by applying an external gradient of a specified value, one can effectively “direct” the nonreactive species to specific locations within the membrane and control the time of the “delivery” of the C component to these locations. Simultaneously, the reactive (AB) parts of the membrane will assume a topology with a characteristic height and depth of the respective AB “hills” and “grooves” that gradually decreases with an increase in the reaction rate coefficient. By applying different values of the external gradient, one can create a range of different topologies within such responsive reactive membrane. Finally, we note that the migration of the nonreactive C component along the gradient is a robust process and depends on the properties of the reactive A and B components. The C component can be any nonreactive impurity that is immiscible with the A and B components. This migration can be the basis for a “self-cleaning” function: the external gradient will effectively wipe off all the impurities and “deliver” them to a designated “collection” region, leaving the rest of the membrane free of any nonreactive immiscible components. Acknowledgment. The authors gratefully acknowledge financial support from the DOE. LA7028615
