Strong bonds and far-from-equilibrium conditions minimize errors in lattice-gas growth
SStrong bonds and far-from-equilibrium conditions minimize errors in lattice-gasgrowth
Stephen Whitelam ∗ Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
We use computer simulation to study the layer-by-layer growth of particle structures in a latticegas, taking the number of incorporated vacancies as a measure of the quality of the grown structure.By exploiting a dynamic scaling relation between structure quality in and out of equilibrium, wedetermine that the best quality of structure is obtained, for fixed observation time, with stronginteractions and far-from-equilibrium growth conditions. This result contrasts with the usual as-sumption that weak interactions and mild nonequilibrium conditions are the best way to minimizeerrors during assembly.
Introduction –
Molecular self-assembly is usually doneusing interaction strengths (cid:15) comparable to the ther-mal energy k B T (henceforth set to unity) and small val-ues of the bulk free-energy difference ∆ g between thestructure and the parent phase [1–6]. Small values of (cid:15) , proportional to the logarithm of the microscopic re-laxation time, allow particles to unbind and correct er-rors during assembly [7–11]. Small values of ∆ g result inslow growth, allowing more time for this error-correctionmechanism to operate. Such mild conditions thereforeseem a natural choice for minimizing errors during as-sembly. Here we show that this expectation is not trueof layer-by-layer growth in a three-dimensional (3D) lat-tice gas, when the vacancy density φ is used as a mea-sure of the quality of the grown structure. We find that φ obeys a scaling relationship ( φ − φ eq ) /φ eq ∝ τ r /τ g ,which contains the equilibrium vacancy density φ eq , andthe ratio of the growth timescale τ g and the microscopicrelaxation timescale τ r . For fixed observation time, thehighest-quality structures – i.e. those with fewest vacan-cies – are made by using large values of (cid:15) and ∆ g , andare out of equilibrium.This prescription results from a competition betweenthe thermodynamic and dynamic factors present in thescaling relation. Large (cid:15) favors few vacancies for two rea-sons. First, the smallest achievable value of φ is the equi-librium vacancy density, φ eq , and this decreases expo-nentially with (cid:15) because vacancies are thermally-exciteddefects. Second, grown structures are more likely tobe in equilibrium (for fixed ∆ g ) as (cid:15) increases , becausethe ratio τ r /τ g decreases. This is so because layer-by-layer growth proceeds via successive nucleation of 2Dlayers on a 3D structure [12–18]. The logarithm of thetime for the advance of each layer scales as σ / ∆ g [12](where σ ∼ (cid:15) [19] is the surface tension between struc-ture and environment), and so grows faster with (cid:15) thandoes the logarithm of the microscopic relaxation time.Thus more microscopic binding and unbinding eventstake place during assembly as (cid:15) increases at fixed ∆ g ,and the structure grown is more likely to be in equilib-rium. Set against these two factors, as (cid:15) increases, larger ∗ [email protected] values of ∆ g are required to produce structures on ob-servable timescales, and as ∆ g increases the ratio τ r /τ g increases. For large enough ∆ g we observe the formationof nonequilibrium structures, which contain more vacan-cies than their equilibrium counterparts. However, this isan acceptable compromise: for fixed observation time thehighest-quality structures are obtained for large values of (cid:15) and ∆ g , such that φ eq is small and τ r /τ g = O (1). Thestructure produced under these conditions is a nonequi-librium one, but is of higher quality than any equilibriumstructure that can be grown on comparable timescales.This simple model system therefore defies the expecta-tion that mild nonequilibrium conditions are the best wayto minimize errors during assembly. Model and results –
Consider the 3D Ising latticegas, which has been used extensively to study crystalgrowth [12–16, 20–22]. We consider occupied sites to beparticles, and unoccupied sites to be vacancies. Nearest-neighbor particles receive an energetic reward of − (cid:15) < µ = 3 (cid:15) − ∆ g fora particle relative to a vacancy (in Ising model languagewe have coupling J = (cid:15)/ h = ∆ g/ (cid:15) > .
886 we are in the two-phase region, wherean interface between the particle phase and the vacancyphase is stable [23]. The bulk free-energy difference be-tween particle and vacancy phases, the thermodynamicdriving force for growth, is ∆ g . Note that the drivingforce for growth is independent of (cid:15) : increasing (cid:15) makesparticles ‘stickier’, but also reduces the effective particleconcentration in ‘solution’, ≈ e − (cid:15) +∆ g .We used lattices of L x × L y × L z sites. For most sim-ulations we set L x = L y = 20 and L z = 50. We imposedperiodic boundaries in x - and y dimensions, and closedboundaries in z , which, through choice of initial condi-tions, is the growth direction. We began each simulationwith a layer of particles in the L z = 0 plane, in orderto study growth without having to wait for nucleationof a 3D structure. We evolved the system using a ki-netically constrained grand-canonical Metropolis MonteCarlo algorithm, similar to that used in Refs. [24]. Ateach step we selected at random a lattice site, and pro-posed a change in state of that site. If the chosen site hadfewer than 6 particles as neighbors then we accepted theproposal with probability min(1 , e − ∆ E ), where ∆ E is the a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug (a) (b) ✏ = 3 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ = 3 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − φ .
25 0 . .
75 1∆ µ τ g .
25 0 . .
75 1∆ µ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) FIG. 1. (a) Characteristic time τ g to grow a layer of the particle structure, as a function of ∆ g , for four values of (cid:15) . Overlaiddotted lines are the analytic result (3). Inset: snapshot of a growing structure, at the indicated state point; green particlesare those in the nucleating layer (see Fig. A1 for additional detail). (b) Vacancy density φ as a function of ∆ g for the samefour values of (cid:15) . Upwards-sloping lines with triangle symbols are the values obtained immediately after a structure of 50 layerswas grown. Errorbars are shown sparsely, for clarity. Downwards-sloping lines with square symbols are equilibrium results; forsmall values of φ these results approach the estimate (4) (dashed lines without symbols). change in energy resulting from the proposed move. If thechosen site had 6 particles as neighbors then we rejectedthe move. The purpose of this constraint is to mimic theslow internal relaxation of solid structures: in the ab-sence of the constraint, vacancies internal to the particlestructure can simply fill in, which would not happen ina real growth process. This algorithm and model cap-ture in a simple way some of the key physical features ofgrowth, principally that particles can bind and unbind atthe growth front, but not within a solid structure. To de-termine equilibrium we used a standard grand-canonicalMetropolis Monte Carlo algorithm with no kinetic con-straint. Both constrained and unconstrained algorithmssatisfy detailed balance with respect to the same energyfunction, and so give rise to identical thermodynamics inthe long-time limit.For ∆ g > z -direction. In Fig. 1(a) we show the characteristic timeto grow one lattice site in the z -direction, averaged overseveral independent simulations (of order 10 at the small-est values of ∆ g , and up to 10 at larger values of ∆ g ) inwhich 50 layers were grown. The growth time increaseswith (cid:15) and decreases with ∆ g . When (cid:15) is small ( (cid:46) (cid:15) the growth frontbecomes smooth [12]. This is the layer-by-layer growthregime. Here it is possible to estimate the growth time byapproximating the growth front as a 2D Ising model [12](see Appendix A) and calculating the free-energy barrier(and consequent rate) for the nucleation of successive lay-ers. This can be done analytically using the results ofRyu and Cai [25, 26]. Those authors showed that the free-energy cost G ( N ) for the formation of a cluster ofsize N in the 2D Ising model can be precisely describedby the equation G ( N ) = − hN + b √ N + τ ln N + G , (1)where b ≡ σ √ π and G ≡ J − b . The first term in(1) is the bulk free-energy reward for growing the stablephase. The term in √ N is the cost for creating inter-face between particles and vacancies; σ is the surfacetension [19, 27] [28]. These are the usual terms writ-ten down in classical nucleation theory (CNT) [18]. Theterm logarithmic in N (with τ = 5 / d = 2) canbe interpreted to account for cluster-shape fluctuations.This term is not usually part of a CNT formulation, butis needed to ensure precise agreement with free energiesobtained from umbrella sampling [25, 26, 29]. The term G in Eq. (1) ensures that G (1) = 8 J − h , which is thefree-energy cost for creating one particle in a backgroundof vacancies.The critical cluster size N c is the value of N thatmaximizes Eq. (1), and is N c = C σ π/ (4 h ), where C ≡ (1 + (cid:112) hτ /σ π ) / G max = G ( N c ); (2)see Fig. A2 [30] (in the absence of the logarithmic term, G max − G = σ π/ (2 h ), familiar from CNT). We thenestimate the characteristic time for the advance of the φ ( ) / φ e q − − (a) (b) ✏ = 2 .
55 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − − − P ( A ) . A − − τ r e l / τ g ϵ − − τ r e l / τ g ϵ ⌧ rel ⌧ g (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) (c) ⌧ rel ⌧ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − − − P ( A ) . A ∆ ϵ = 2 . ϵ = 2 . ϵ = 2 . ϵ = 2 . ϵ = 3 ⌧ rel ⌧ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − φ .
25 0 . .
75 1∆ µ φ ( ) / φ e q − − φ / φ e q − − τ rel /τ g ϵ = 3 ϵ = 2 . (a) (b) ✏ = 2 .
55 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − − − P ( A ) . A − − τ r e l / τ g ϵ − − τ r e l / τ g ϵ ⌧ rel ⌧ g (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) (c) ⌧ rel ⌧ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − − − P ( A ) . A ∆ ϵ = 2 . ϵ = 2 . ϵ = 2 . ϵ = 2 . ϵ = 3 ⌧ rel ⌧ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) φ / φ e q − − τ rel /τ g φ ( ) / φ e q − − − − φ .
25 0 . .
75 1∆ µ y1=1.0+7.5*x1; φ ( ) / φ e q − φ / φ e q − − τ rel /τ g g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) ⌧ r /⌧ g (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) FIG. 2. (a) Probability P ( A ) that, during growth, a lattice site has undergone A changes of state after first acquiring 6neighbors. Data are for driving force ∆ g = 0 .
51 for various (cid:15) ; the larger is (cid:15) , the more dynamic are particles on the timescaleof growth. Inset: the molecular relaxation time divided by the growth time decreases with increasing (cid:15) . (b) Dynamic data forlarge (cid:15) , in the format of Fig. 1(b), collapse when rescaled in the manner shown in panel (c). This collapse indicates that grownstructures’ vacancy densities are controlled by the ratio of relaxation and growth timescales. Inset: the vacancy density φ (0)of the fresh bulk, scaled by the bulk equilibrium value φ eq , is also a function of τ r /τ g . growth front to be τ g = τ exp( G max ) , (3)for sufficiently large G max , where τ = 10 − is a constantthat we determined by comparison with simulation. Theestimate (3) agrees with the simulation data of Fig. 1for sufficiently large (cid:15) and sufficiently small ∆ g . Thiscomparison confirms that growth in the regime (cid:15) (cid:38) (cid:15) for arbitrarily large values of thatparameter.We next assess how close to equilibrium is the struc-ture produced immediately after the growth process. InFig. 1(b) we show the vacancy density φ , the numberof vacancies divided by the total number of sites, withinthe middle 50% of the simulation box (between the planes z = L z / z = 3 L z /
4) immediately upon completionof layer L z = 50. For comparison we show the valueof φ in equilibrium, φ eq . For small values of φ eq theseequilibrium values approach the estimate φ (0)eq = (cid:0) (cid:15) +∆ g (cid:1) − ; (4)note that 6 (cid:15) − µ = 3 (cid:15) +∆ g is the energy cost for removinga particle from the bulk of a vacancy-free structure.Comparison of growth and equilibrium results indi-cates that, for all values of (cid:15) studied, there exists for suf-ficiently small ∆ g a ‘quasiequilibrium’ regime [9, 11, 31].Here the initial outcome of growth is the equilibriumstructure. The vacancy density of sites that have justacquired 6 neighbors for the first time, which we callthe fresh bulk, is not the bulk equilibrium value (seeFig. A3(a)). However, particles in the growth front can unbind, leaving a temporary hole (Fig. A1(a)), and al-lowing sites in the fresh bulk to change state. For small∆ g , such processes occur enough times that the layersadjacent to the growth front equilibrate before the frontmoves away.By contrast, for larger values of ∆ g the bulk of thestructure is not in equilibrium. The fresh bulk failsto equilibrate in the presence of the growth front, andbecomes trapped out of equilibrium as the front movesaway. The timescale for subsequent relaxation to equi-librium is very long, because vacancies, which effectivelymove by diffusion (see Appendix B), cannot catch theballistically-moving growth front. Nonequilibrium trap-ping of impurities [32, 33] and vacancies [34] is seen incrystal growth. Notably, for (cid:15) (cid:38)
2, the value of ∆ g atwhich the grown structure falls out of equilibrium in-creases with increasing (cid:15) : ‘colder’ structures are betterequilibrated. To understand this result, recall that thegrowth time scales approximately as the exponential ofthe free-energy barrier to layer nucleation, or approxi-mately as the exponential of (cid:15) . By contrast, we esti-mate the microscopic relaxation time τ r at the growthfront (or in the bulk next to a vacancy) to be the charac-teristic time required to remove a particle with 5 bonds.The energy cost for doing so is 5 (cid:15) − µ = 2 (cid:15) + ∆ g , and sowe estimate τ r = e (cid:15) +∆ g . (5)Thus the growth time increases faster with (cid:15) than doesthe relaxation time, and so more molecular relaxationevents take place during growth at large (cid:15) ; see Fig. 2(a).We can justify the estimate (5) for relaxation time byrescaling the (cid:15) (cid:38) φ that differ by about anorder of magnitude, and growth times τ g that differ byseveral orders of magnitude. Such dynamic scaling is alsoseen in simulations of crystal growth in the presence ofimpurities [20, 21, 33], vapor deposition of glasses [35],irreversible polymerization [36], and the growth of model1D structures [37].The black dotted line in Fig. 2(c) has equation φ = φ eq (cid:18) k τ r τ g (cid:19) , (6)with k = 0 .
15. This expression emphasizes that theoutcome of self-assembly is a combination of thermody- τ g µ − − φ µ slow growth slow relaxationQE limit (a)(b) µ ϵ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − − − φ ϵ QE τ g ≤ τ g ≤ τ g ≤ τ g ≤ τ g ≤ − − − − − φ ϵ QE τ g ≤ τ g ≤ τ g ≤ τ g ≤ τ g ≤ FIG. 3. (a) The scaling relation (6) can be used to extrapo-late to lengthscales and timescales beyond the reach of sim-ulation (results are for (cid:15) = 4; QE denotes ‘quasiequilibrium’,where the initial outcome of growth is the equilibrium stru-ture). (b) Eq. (6) can also be used to determine the protocolthat minimizes φ . Each line shows the smallest φ accessible,as a function of (cid:15) , for given observation time (we terminatelines when τ r /τ g exceeds 20). The inset shows the value of∆ g required to produce each structure. For each choice ofobservation time, φ is minimized for large (cid:15) and ∆ g , and thestructure grown is not an equilibrium one. namics and dynamics. It also shows how a quasiequilib-rium regime, for which φ = φ eq , emerges when driving isweak. As ∆ g is made small, the growth time diverges – itscales to leading order as exp( σ π/ ∆ g ) – while the molec-ular relaxation time approaches a constant, rendering τ r /τ g ≈
0. By contrast, for the growing two-componentfiber of Ref. [37] there is no quasiequilibrium regime, be-cause growth and relaxation times remain strongly cou-pled even for weak driving. These distinct behaviors in-dicate an important difference between growth processesin 1D and 3D.For large (cid:15) the quantities φ eq , τ g and τ r are accuratelydescribed by Equations (4), (3), and (5), respectively, andin that regime we can use (6) to extrapolate analyticallythe data of Fig. 1 to lengthscales φ − / and timescales τ g beyond those accessible to simulation: see Fig. 3(a) andFig. A4. We can also use it to determine the protocolfor producing the highest-quality structure. In Fig. 3(b)we show the smallest value of φ , as a function of (cid:15) , ac-cessible on a given observation time (the inset shows thecorresponding value of ∆ g ). In all cases φ is minimizedby large values of (cid:15) and values of ∆ g large enough thatthe structure grown is a nonequilibrium one. In essence,the prescription for the best-quality structure is to have (cid:15) large, so that φ eq is small, and drive the system hardso that the structure grows on the accessible timescale.Consequently, τ r /τ g (cid:38)
1, meaning that growth is far fromequilibrium and results in a nonequilibrium structure.
Conclusions –
The majority of self-assembled materi-als made with few defects are prepared using weak in-teractions and mild nonequilibrium conditions, but wehave shown that vacancy incorporation in the layer-by-layer growth of a 3D lattice gas is minimized using stronginteractions and far-from-equilibrium conditions. Find-ing error-minimization protocols is important for the as-sembly of certain types of nanomaterials. For instance,DNA bricks are distinguishable structures built from Q ‘brick’ types, in which each brick possesses a defined loca-tion [1, 38]. The interaction energies of bricks must growas (cid:15) ∼ ln Q in order to thermally stabilize the assembly(to counter the entropy of permutation ln Q ! possessedby disordered arrangements of bricks). The present worksuggests one way to incorporate strong interactions intoa productive assembly protocol. Acknowledgments –
I thank Jeremy D. Schmit forvaluable discussions and comments on the manuscript.This work was done at the Molecular Foundry, LawrenceBerkeley National Laboratory, and was supported by theOffice of Science, Office of Basic Energy Sciences, of theU.S. Department of Energy under Contract No. DE-AC02–05CH11231. [1] A. Reinhardt and D. Frenkel, Physical Review Letters , 238103 (2014).[2] Z. Zhang and S. C. Glotzer, Nano Letters , 1407 (2004).[3] E. Bianchi, P. Tartaglia, E. La Nave, and F. Sciortino,The Journal of Physical Chemistry B , 11765 (2007).[4] D. Nykypanchuk, M. M. Maye, D. Van Der Lelie, andO. Gang, Nature , 549 (2008).[5] G. M. Whitesides and B. Grzybowski, Science , 2418(2002).[6] M.-P. Valignat, O. Theodoly, J. C. Crocker, W. B. Rus-sel, and P. M. Chaikin, Proceedings of the NationalAcademy of Sciences of the United States of America , 4225 (2005).[7] M. F. Hagan and D. Chandler, Biophysical Journal ,42 (2006).[8] A. W. Wilber, J. P. Doye, A. A. Louis, E. G. Noya, M. A.Miller, and P. Wong, The Journal of Chemical Physics , 085106 (2007).[9] D. Rapaport, Physical Review Letters , 186101(2008).[10] M. Hagan, O. Elrad, and R. Jack, The Journal of Chem-ical Physics , 104115 (2011).[11] S. Whitelam and R. L. Jack, Annual Review of physicalchemistry , 143 (2015).[12] W.-K. Burton, N. Cabrera, and F. Frank, PhilosophicalTransactions of the Royal Society of London A: Math-ematical, Physical and Engineering Sciences , 299(1951).[13] G. Gilmer, Journal of Crystal Growth , 15 (1976).[14] G. Gilmer, Science , 355 (1980).[15] K. A. Jackson, Kinetic Processes: Crystal Growth, Dif-fusion, and Phase Transformations in Materials (JohnWiley & Sons, 2006).[16] J. D. Weeks and G. H. Gilmer, Adv. Chem. Phys , 157(1979).[17] J. J. De Yoreo and P. G. Vekilov, Reviews in mineralogyand geochemistry , 57 (2003).[18] R. P. Sear, Journal of Physics: Condensed Matter ,033101 (2007).[19] L. Onsager, Physical Review , 117 (1944).[20] K. A. Jackson, Interface Science , 159 (2002).[21] K. A. Jackson, K. M. Beatty, and K. A. Gudgel, Journalof Crystal Growth , 481 (2004).[22] K. A. Jackson, G. H. Gilmer, and D. E. Temkin, PhysicalReview Letters , 2530 (1995).[23] G. Pawley, R. Swendsen, D. Wallace, and K. Wilson,Physical Review B , 4030 (1984).[24] S. Whitelam, L. O. Hedges, and J. D. Schmit, PhysicalReview Letters , 155504 (2014).[25] S. Ryu and W. Cai, Physical Review E , 011603 (2010).[26] S. Ryu and W. Cai, Physical Review E , 030601 (2010).[27] V. A. Shneidman, K. A. Jackson, and K. M. Beatty, TheJournal of Chemical Physics , 6932 (1999).[28] Here σ ≡ ( σ (cid:107) + σ diag ) / (2 √ χ ), with σ (cid:107) ≡ J − ln coth J , σ diag ≡ √ J , and χ ≡ (1 − sinh − J ) / .[29] L. O. Hedges and S. Whitelam, Soft Matter , 8624(2012).[30] If the cross-sectional area N ⊥ ≡ N x × N y of the simulationbox is too small to accommodate the 2D critical cluster, N ⊥ < N c , then (2) should be replaced by G ( N ⊥ ).[31] R. L. Jack, M. F. Hagan, and D. Chandler, Physical Review E , 021119 (2007).[32] H. J. Leamy, J. C. Bean, J. Poate, and G. Celler, Journalof Crystal Growth , 379 (1980).[33] A. Kim, R. Scarlett, P. Biancaniello, T. Sinno, andJ. Crocker, Nature materials , 52 (2008).[34] C. D. Van Siclen and W. Wolfer, Acta metallurgica etmaterialia , 2091 (1992).[35] L. Berthier, P. Charbonneau, E. Flenner, and F. Zam-poni, arXiv preprint arXiv:1706.02738 (2017).[36] S. Corezzi, C. De Michele, E. Zaccarelli, P. Tartaglia,and F. Sciortino, The Journal of Physical Chemistry B , 1233 (2009).[37] S. Whitelam, R. Schulman, and L. Hedges, Physical Re-view Letters , 265506 (2012).[38] Y. Ke, L. L. Ong, W. M. Shih, and P. Yin, Science ,1177 (2012).[39] M. Hasenbusch, S. Meyer, and M. P¨utz, Journal of sta-tistical physics , 383 (1996).[40] G. H. Fredrickson and H. C. Andersen, Physical ReviewLetters , 1244 (1984). Appendix A: Approximation of the growth front as a 2D Ising model
For (cid:15) (cid:38) .
630 the equilibrium interface between particles and vacancies is statistically smooth [39]. For sufficientlylarge (cid:15) ( (cid:38)
2) it is convenient to consider the exposed surface of a particle structure growing in the z -direction tobe a two-dimensional (2D) Ising model [12]. If the layer adjacent to the exposed surface has no vacancies then theexposed layer behaves as a 2D Ising model whose parameters are the same as the 3D Ising model given in the maintext, J = (cid:15)/ h = ∆ g/
2. To see this, note that the Hamiltonian of the exposed layer is H = − (cid:15) (cid:88)
762 [19]. Thus if we approximate the surface of thestructure as a 2D Ising model, then, for (cid:15) (cid:38) . Appendix B: Internal relaxation of the bulk
Vacancies trapped within the structure can undergo diffusion, in an effective way, even in the presence of the kineticconstraint: the particle adjacent to the vacancy, which has fewer than 6 neighbors, can become a vacancy, and thenthe original vacancy can be filled in. In addition, two vacancies that meet each other can coalesce, leaving behindonly a single vacancy. This internal dynamics of (effective) vacancy diffusion and coalescence is similar to that ofspins in the kinetically constrained Fredrickson-Andersen model [40]. Vacancy coalescence can lead to evolution ofthe bulk structure toward the equilibrium vacancy density, which we see for sufficiently small values of (cid:15) ( (cid:46) (cid:15) (cid:38)
2, the vacancy density φ is independent of observation time, for the range of timesstudied, showing that no aging of the structure has occurred on the growth timescale. Thus the dynamically-generatedvacancy density results only from dynamics that occurs in the presence of the growth front, and not from subsequentrelaxation of the bulk of the structure. Vacancy coalescence is unphysical in the sense that it would not happen withinthe bulk of a solid structure, and so we focus our attention on the regime of parameter space in which this processdoes not occur. Appendix C: Additional figures G ( N ) 0 100 200 300 N ∆ µ = 0 . µ = 0 . µ = 0 . G ( N ) 0 100 200 300 N ∆ µ = 0 . µ = 0 . µ = 0 . g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) L (1) D e↵ / V (2) D e↵ / D (3) ✓ (4) J ⇠ ln D e↵ (5) m (6) (7) (8) (a) eps=2.55 G ( N ) 0 100 200 300 N ∆ µ = 0 . (b) (c) N c . ∆ µ τ . . . . . µ N c . ∆ µ (d) t (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) L × × t τ g .
25 0 . .
75 1∆ µ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) FIG. A1. Layer-by-layer growth in the 3D lattice gas. (a) Time-ordered configurations of a simulation box of 20 × × (cid:15) = 2 . , ∆ g = 0 .
25 (see e.g. Refs. [13, 14] for similar pictures). L is the number of layers grownin the z -direction; periodic boundaries are applied in the direction perpendicular to growth. Particles are shown blue, with thefollowing exceptions: particles in the nucleating layer are shown green; under-coordinated particles in the layer below that areshown pink; and the exposed particles in the layer below that are shown yellow. The first snapshot shows a critical 2D cluster(green). (b) A plot of L versus t shows that the interface pauses, for varying amounts of time (number of Monte Carlo sweeps),between 2D nucleation events. (c) These nucleation events are governed by free-energy profiles, Eq. (1), for 2D clusters on thesurface of the 3D structure. We show such profiles, as a function of cluster size N , for three values of ∆ g . These profiles assumethat the layer adjacent to the nucleating layer is defect-free, which is approximately true for large (cid:15) . (d) Characteristic growthtime τ g as a function of ∆ g (blue line). Overlaid as a dashed line is Eq. (3), showing that, for sufficiently small ∆ g , the scalingof growth time follows from consideration of 2D nucleation events. The inset shows the size N c of the 2D critical cluster as afunction of ∆ g . ✏ (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4) ⇢ ( m ) ⇠ e KI ( m ) (5) I coin = N ln ⇢ ( b/N , N ) = N ln 2 N ✓ Nb ◆ (6) P (ref ) [ x ] / K Y k =1 p (ref ) ( C k ! C k +1 ) (7) I ( m ) / m (8) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (9) N = b + r (10)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (11) J ⇠ ln D e↵ (12) m (13) (14) ✏ (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4) ⇢ ( m ) ⇠ e KI ( m ) (5) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (6) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (7) I ( m ) / m (8) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (9) N = b + r (10)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (11) J ⇠ ln D e↵ (12) m (13) (14) ✏ (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) . . . . . . . g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) FIG. A2. Free-energy barrier, Eq. (2), to nucleation of a 2D layer on the surface of a defect-free 3D structure. The barrierincreases with decreasing ∆ g or increasing (cid:15) . Its increase with (cid:15) is approximately quadratic. Thus the growth time of the3D structure, roughly the exponential of the barrier, grows faster with (cid:15) than the molecular relaxation time, which scalesexponentially with (cid:15) . (a) (b) − − − − − P ( A ) . A ∆ µ = 0 . µ = 0 . µ = 0 . − − − − − P ( A ) . A ∆ µ = 0 . µ = 0 . µ = 0 . − − − − φ .
25 0 . .
75 1∆ µ ✏ = 3 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) ✏ = 2 .
55 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) − − − φ .
25 0 . .
75 1∆ µ ✏ = 3 (1) µ (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)0 k B T (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) (c) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) FIG. A3. (a) As Fig. 1(b), but including data (upwards-sloping lines with circles) indicating the vacancy density of the ‘freshbulk’, i.e. the fraction of sites that are vacant upon first acquiring 6 neighbors. (b) Probability distribution P ( A ) of the numberof times A that a site changes state after first acquiring 6 neighbors, for three values of ∆ g and for (cid:15) = 2 .
55. Such changes ofstate allow the ‘fresh bulk’ adjacent to the growth front to evolve into the ‘mature bulk’ (triangle symbols in panel (a)). Forsmall values of ∆ g such evolution is sufficient to attain equilibrium while a site is close to the growth front; for large valuesof ∆ g it is not. (Here and in Fig. 2 we show even values of A ; histograms for odd values of A show similar behavior.) (c)As Fig. 1(b) but with additional dynamic data: light blue and orange lines show the vacancy density φ immediately after thegrowth of 25 and 100 layers, respectively (the data of Fig. 1(b) are obtained immediately after the growth of 50 layers). Forthe cases (cid:15) = 1 . (a) (b) τ g µ ϵ = 3 ϵ = 4 ϵ = 5 ϵ = 6 − − − φ µ g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) g (1) (2) N ⌘ r + b (3) m ⌘ ( b r ) /N (4)3 (5)20 k B T (6)40 k B T (7) I coin = N ln ⇢ ( b/N, N ) = N ln 2 N ✓ Nb ◆ (8) P (ref) [ x ] / K Y k =1 p (ref) ( C k ! C k +1 ) (9) I ( m ) / m (10) m ? = ˙ b ( m ? ) ˙ r ( m ? )˙ b ( m ? ) + ˙ r ( m ? ) (11) N = b + r (12)˙ N = 0 = ) ˙ b ( m ) + ˙ r ( m ) = 0 (13) J ⇠ ln D e↵ (14) FIG. A4. As Fig. 1, but extrapolated to larger values of (cid:15) (longer times and smaller impurity densities) using Eq. (6). For (cid:15)(cid:15)