Supporting information for Self-assembly of multicomponent structures in and out of equilibrium

Transcription

1 Supporting inforation for Self-assebly of ulticoponent structures in and out of equilibriu Stephen Whitela 1, Rebecca Schulan 2, Lester Hedges 1 1 Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 9472, USA 2 Cheical and Bioolecular Engineering, Johns Hopkins University, 34 N Charles St, Baltiore, MD 21218, USA I. MOTIVATION OF ANALYTIC THEORY The analytic theory developed in the ain text is a dynaical ean-field one, which focuses on the likelihood φ [(η i + 1)/2] that a given fiber bond is a bulk one. Here η i S i S i+1, is a defect variable : η i = 1 describes a bulk (sae-color) bond, and η i = 1 is a defect (unlike-color) one. We expect a ean-field theory of this nature to be ost accurate when red and blue blocks are added to the fiber with equal likelihood, because, in this liit, the Ising odel representation of fiber energetics can be written in the noninteracting (space-independent) for H = J i η i. To derive Eqns. and of the ain text, we argue as follows. At any instant, the end of the fiber is either red or blue. Regardless, the next block added is of atching color with probability 1/2, and so we expect bulk doains to grow with rate c/2. We expect bulk doains to shrink with rate φe βɛs : the factor φ ensures that the terinal bond is a bulk one, and the factor e βɛs contains the energy scale for the bulk interaction. Siilarly, we expect the fiber as a whole to grow with rate c, and to shrink with rate e βɛs (resp. e βɛ d ) if its rightost bond is a bulk (resp. defect) one. These arguents iply the drift velocities for bulk doains and for the fiber given in Eqns. and of the ain text. II. DYNAMIC CORRELATION LENGTH The dynaic correlation length ξ derived fro Eq. of the ain text is 2(e βɛs e βɛ d ) ξ(c, ɛ s, ɛ d ) = e βɛs + e βɛ d (c e βɛ s 1) (e βɛ d e βɛ s) 2 + c 2 e 2β(ɛs+ɛ d) (S1) when ɛ s ɛ d, and ξ = 2 otherwise. swhitela@lbl.gov rschul3@jhu.edu

2 2 III. DYNAMICAL MEAN-FIELD THEORY FOR HIGHER DIMENSIONS Dynaical ean-field theory suggests that the far-fro-equilibriu assebly strategy otivated by our fiber work is transferrable to higher diensions. One possible higher-diensional generalization of the fiber odel is a collection of utually attractive red and blue blocks diffusing in (d > 1)-diensional solution (see e.g. Ref. [25] of the ain text). Post-nucleation, we study the growth of a structure coposed of red and blue blocks by following the otion of the interface or front between the structure and solution (see Fig. S6(a)). We can consider such a growth process within ean-field theory. We assue that blue and red blocks are added with equal rates c/2 to the front, whereupon each block akes z energetic bonds with the front (as an approxiation we fix z to be constant, unaffected by binding events). In units of k B T the red-red and blue-blue interactions are both ɛ s, while the red-blue interaction is ɛ d. We allow the coposition of the surface to be described by a paraeter [ 1, 1], siilar to an Ising agnetization: when = 1 the front is all red; when = 1 the front is blue; and when = the front is randoly ixed. Each of the z available surface bonds is then red with probability p R = (1 )/2, and blue with probability p B = 1 p R. The energies per bond felt by red and blue blocks in a front of coposition are ɛ red () = ɛ s p R ɛ d p B = 1 2 (ɛ s + ɛ d ) + 2 (ɛ s ɛ d ), (S2) and ɛ blue () = ɛ s p B ɛ d p R = 1 2 (ɛ s + ɛ d ) 2 (ɛ s ɛ d ). (S3) Consideration of possible binding and unbinding events iplies that the net rates of attachent of red and blue blocks are respectively Γ R = c/2 p R exp (βz ɛ red ()) = c/2 1 (1 ) exp ( Σ + ), (S4) 2 and Γ B = c/2 p B exp (βz ɛ blue ()) = c/2 1 (1 + ) exp ( Σ ), (S5) 2 where Σ βz(ɛ s + ɛ d )/2 and βz(ɛ s ɛ d )/2. The key difference fro the one-diensional case is that the energies felt by red and blue blocks depend in a continuous way on the state (redness or blueness) of the growing front. This dependence introduces a cooperatively not present in one diension. In equilibriu, i.e when Γ R = Γ B =, Equations (S4) and (S5) give a self-consistent equation for the equilibriu coposition : ( ) βz = tanh 2 (ɛ s ɛ d ). (S6) This equation resebles the failiar Ising odel equation of state in zero field, and describes a phase transition between a high-teperature ixed red/blue phase ( = ), and a low-teperature phase of coexisting red ( < ) and blue ( > ) doains (see Fig. S6). The color syetry of the front is therefore spontaneously broken. The presence of a therodynaic phase transition at finite teperature, a fundaental property of 2- and 3-diensional systes generally, constitutes a key difference fro the 1d fiber odel. A siilar color-deixing transition is seen in our siulations of two-color square-well-attractive particles arranged on an fcc lattice (see Fig. S6). Although the critical exponents associated with color deixing in 3d are not the values predicted by ean-field theory, the qualitative behavior is siilar. Significantly, this higher-diensional odel possesses scaling properties siilar to those that allow the far-fro equilibriu nonperturbative assebly strategy described in the ain text (for d = 1). To solve the present odel s

3 dynaics, we ipose the self-consistent condition Γ R /Γ B = p R /p B, i.e. the requireent that the relative net attachent rate of red and blue blocks is equal to the relative fraction of red and blue blocks in the growing front. Equations (S4) and (S5) inserted into this condition give the self-consistent equation = c + c 2 + 4e 2Σ sinh 2 ( ) 2e Σ. (S7) sinh( ) Eq. (S7), a dynaical generalization of Eq. (S6), describes a dynaic phase transition between ixed and de-ixed phases, with a critical teperature that depends on the growth speed of the front. In Fig. S6 we show dynaic phase behavior for concentrations c twice- and six ties the value at the phase boundary, c (T ) (found by inserting, obtained fro Eq. (S6), into Eq. (S4), with Γ R = ). Note that the concentration at the phase boundary is teperature-dependent. As in one diension, structures produced at finite rate of growth are invariably not the equilibriu one, and are ore ixed than their equilibriu counterpart. Iportantly, though, Eq. (S7) suggests that the strategy otivated by our fiber work is transferable to higher diensions. In Fig. S6 we show contours of constant front coposition, as a function of concentration c and the energetic paraeter ɛ s. This plot is siilar to Fig. 2 of the ain text in the sense that structures of given coposition can be generated at and away fro equilibriu. We expect the d > 1 ean-field odel developed here to describe qualitatively the color patterns within growing structures coprised of siple two-color building blocks in 3d. The two-color square-well particles used to generate Fig. S6 fit this description, for exaple. Concretely, Fig. S6 predicts that increasing the value of the like-color building block interaction alone will result in a growing front that is ore ixed than the equilibriu structure, because increasing binding energy increases growth front velocity v(c, ) = Γ R +Γ B (which can be straightforwardly calculated fro Eqns. (S4) and Eq. (S5)), and so increases the degree of ixing of the front. But by increasing the like-color interaction and decreasing particle concentration according to the prescription described by a given contour, structure can be aintained. Siulations or experients are now required to test whether such pattern control can be exerted in a regie in which structures reain orphologically ordered. 3

4 4 IV. SUPPLEMENTAL FIGURES Figs. S1 S4 suppleent Fig. 1 of the ain text; Fig. S5 suppleents Fig. 2 of the ain text. Fig. S6 illustrates the behavior of the d > 1 dynaical ean-field odel described in this suppleent.

5 5 ρ(l) c = 1 s =3, d =1 c =2.37 c c =1.34 c irrev. eq. noneq. c = c l FIG. S1: Dynaic fiber doain length distributions are exponential. We show ρ(l) obtained fro siulations (grey lines), for selected concentrations at and past the phase boundary c = c, for a set of conditions considered in Fig. 1 of the ain text. Overlaid are analytic estiates of the doain length distribution, ρ(l) = (ξ 1) 1 exp [ l ln(1 ξ 1 ) ], where the ean doain length ξ is obtained fro self-consistent ean field theory (Eq. (S1)) (blue). When c = c this reduces to the equilibriu value ξ = 1 + exp(β(ɛ s ɛ d )) (green); when c we obtain the rando adsorption liit ξ = 2. In all cases the analytic expressions atch the siulation results. Because observed doain length distributions are exponential, we consider only the ean doain length ξ in Fig. 1 of the ain text.

6 6 δf c s FIG. S2: Excess free energy δf f(φ) f(φ ) per fiber block as a function of concentration c and like-color energy scale ɛ s (note that ɛ d = 1). Here φ is the equilibriu bulk fiber fraction, and φ is its dynaic counterpart, here coputed using Eq. of the ain text. The free energy is f(φ) = (ɛ s ɛ d )(1 φ) + k BT (φ ln φ + (1 φ) ln(1 φ)); at all points past the phase boundary, dynaically-generated structures lie higher in free energy than equilibriu ones. An exception occurs when ɛ s = ɛ d, because there the copositional correlations of the equilibriu structure are equal to those of rando ixing.

7 7 1.5 ɛ d =4 ɛ d =3 analytic ξ c FIG. S3: As Fig. 1 of ain text, but for the case ɛ d > ɛ s (here ɛ s = 1), giving an effective Ising antiferroagnetic coupling J = (ɛ s ɛ d )/2 <. In this case the equilibriu structure iics that of a binary crystal, consisting (in the liit of large ɛ d ) of alternating blue and red blocks. We observe the sae breakdown of the quasiequilibriu assuption as for the case J > (phase boundaries and equilibriu correlation lengths are labeled in the sae anner). However, because the nuerical difference between the doain length associated with rando ixing (ξ = 2) and equilibriu (ξ 1 for large ɛ d ) is sall, structures grown close to the phase boundary are nuerically siilar to their equilibriu counterparts.

8 8 1 analytic siulation.5 c =1 2 analytic siulation s FIG. S4: For fixed absolute supersaturation δc c c, the distance (ξ ξ)/ξ fro equilibriu of dynaically generated fiber structures grows sharply with increasing energy scale ɛ s. Here ɛ d = 1.

9 9 Σ/ ξ ɛs ξ target =15 ξ target =5 ξ target = ɛs FIG. S5: Siulations perfored at the locations specified by the circles in Fig. 2, ain text, bear out the analytic prediction that identical structures are generated along contours (top panel), by growth protocols displaying arkedly different degrees of icroscopic reversibility (botto panel). Σ N + + N is the total nuber of binding (N +) and unbinding (N ) events taking place during the assebly of fibers of length L = blocks ( N + N = L is the net nuber of binding events). At the phase boundary, fibers grow only diffusively. As a result, L 2 icroscopic events are required to generate a structure of length L. Far fro the phase boundary, assebly is uch less reversible.

10 c = eq. 4c(T ) growing c = (a) 2c(T ) front 1/( s d) (a) equil. (11) =.2 (11) bo un c e da T = ( s.9 as 1/J d)a c = 2c(T ).4 = ).6 c = 2c(T = c = 2c(T ) 6 c = 4c(T ).94c(Tc.8.7).6 c= ) = 2c(T.5.4 (a).5 c = 4c(T ).2 c) = 2c(T.9 (a) c = ) 4c= (T 3 red blue ) c = 4c(T (a) (a) ns/sec (a) s 1 ns/sec 1 ns/sec developed in the 1paper. ns/sec FIG. S6: Extension to higher diensions of the self-assebly strategy (a) Scheatic of ean-field Eqns. (S6) and analog of fiber odel for d > 1 (see text). Equilibriu and nonequilibriu coposition diagras, fro ry ph c = 2c(T.6 ) 1 ns/sec ixed phase equil. c = 4c(T ) c = 6c(T ) equil. eq..4 eq. 1 ns/secc = 2c (T ) 2c(T c c== 2c (T) ) (a) (a)t c = 2c(T ) 4c(T ) czcc== (T) ) = 4c 4c(T c = 4c(T ).2 Z Z = ln dr exp( U (r)) z binding Z = ln dr(a) exp( blue U (r)) red sites available (a) ZZ =.2 1 ns/sec Z-1Zzzzz == -.5 exp(.5 1 ln dr Uzz(r)) front characterized by 2 [ 1, 1] ln dr exp( Uzz(r)) 8 K =.2 redblue 1 ns/sec 8 K 1 ns/sec 1 ns/sec ixed phase =.4 1 ns/sec = equil. 1 d ) 1 1 ns/sec (S7). These diagras describe a phase transition between a ixed-color structure at high teperature, and phase-separated red and blue doains at low teperature. In the dynaical case, the critical teperature is a function of distance past the siilar equilibriu in concentration phase boundary. Paraeters: s = 2, d = 1, βz = 1/T. Qualitatively deixing is seen siulations: we show coposition diagra (binodal) derived fro Monte Carlo 1siulations of two-color square-well particles T = ( s d) (11) in 3d, which are arranged on a perfect fcc crystal lattice. Particle interaction energies, in units of k T, are s (like colors) and T = ( s d) 1 B (11) d (unlike colors). The deixing transition is qualitatively siilar to the ean-field equilibriu behavior of panel. Mean-field higher-diensional analog of Fig. 2 (ain text). We show contours of coposition, as a function of concentration c and the energetic paraeter s (here d = 1 and βz = 1). This plot is siilar to Fig. 2 of the ain text in the sense that T = ( s boundary), d) 1 suggesting that (11)the structures of given coposition can be generated at and away fro equilibriu (the phase strategy otivated by our fiber work is transferrable to higher diensions. 1 T = ( s (11) =.2 T = ( s d ) (11)