Design Principles for Fast and Efficient Self-Assembly Processes

Self-assembly is a fundamental concept in biology and of significant interest to nanotechnology. Significant progress has been made in characterizing and controlling the properties of the resulting structures, both experimentally and theoretically. However, much less is known about kinetic constraints and determinants of dynamical properties like time efficiency, although these constraints can become severe limiting factors of self-assembly processes. Here, we investigate how the time efficiency and other dynamical properties of reversible self-assembly depend on the morphology (shape) of the building blocks for systems in which the binding energy between the constituents is large. As paradigmatic examples, we stochastically simulate the self-assembly of constituents with triangular, square, and hexagonal morphology into two-dimensional structures of a specified size. We find that the constituents’ morphology critically determines the assembly time and how it scales with the size of the target structure. Our analysis reveals three key structural parameters defined by the morphology: the nucleation size and attachment order, which describe the effective order of the chemical reactions by which clusters nucleate and grow, respectively, and the growth exponent, which determines how the growth rate of an emerging structure scales with its size. Using this characterization, we formulate an effective theory of the self-assembly kinetics, which we show exhibits an inherent scale invariance. This allows us to identify general scaling laws that describe the minimal assembly time as a function of the size of the target structure. We show how these insights on the kinetics of self-assembly processes can be used to design assembly schemes that could significantly increase the time efficiency and robustness of artificial self-assembly processes.

Popular Summary

In biology and nanotechnology, tiny building blocks can quickly and robustly come together to form complex structures. In our study, we dive into this world of self-assembly and uncover a crucial factor: morphology, or the shape of these building blocks. While the importance of weak and reversible interactions between building blocks is well understood, our research sheds new light on how their shapes significantly influence the efficiency of self-assembly.

By simulating self-assembly processes using triangular, square, and hexagonal building blocks, we find that the morphology of the building blocks dictates the speed and efficiency of their self-assembly. Hexagonal blocks, for instance, assemble particularly fast, while other shapes may require significantly more time to form the final structures. To understand this phenomenon mathematically, we develop a model that reveals an inherent scaling symmetry. This symmetry allows us to determine how the assembly time scales as a function of the structure size, thereby explaining the vast differences in time efficiency resulting from different monomer morphologies.

By understanding how morphology impacts assembly efficiency, we can design better self-assembly strategies for various applications. For instance, one might guide the assembly process to form intermediate structures with optimal morphologies, ultimately speeding up the overall assembly process.

Looking ahead, our research opens doors to further exploration. For example, we aim to extend our analysis to investigate how spatial arrangements and assembly errors might affect the efficiency and robustness of self-assembly. Our goal is to open up new avenues for the development of efficient and resilient self-organizing systems through a deeper exploration of the dynamics of self-assembly.

Figure 1

Illustration of the model. (a) We consider the self-assembly of two-dimensional structures of volume (total number of constituents) $S$ from square-, triangle-, or hexagon-shaped building blocks and ask for the time the assembly processes take to reach substantial yields. The configurations shaded in dark blue, red, and yellow illustrate the first intermediate assembly states with enhanced stability (wherein each particle has at least two bonds). Typical assembly pathways consecutively pass through these (or equivalent) intermediate states. The dynamics can, thus, be effectively described by the three parameters $\sigma$, $\gamma$, and $\omega$, which are listed for the three cases and denote the nucleation size, attachment order, and growth exponent, respectively [see (c) and main text Secs. 3a and 4]. (b) Self-assembly dynamics is modeled as follows (illustrated for the system with square-shaped monomers): Two monomers dimerize at rate $\mu$ along any of their edges. Subsequently, monomers attach at rate $\nu$ to any free neighboring site until a square consisting of $S$ monomers with edge length $L=\sqrt{S}$ ($L=\sqrt{S/2}$ in the case of triangular monomers) is completed as illustrated in (a), assuming periodic boundary conditions between top and bottom and left and right. At the same time, particles can also detach from a given structure at rate ${\delta }_n\sim \exp [-n{E}_B/k_{B}T]$, with $n$ denoting the number of bonds between the particle and the structure and $E_B$ the binding energy per bond. (c) Coarse-grained effective dynamics (illustrated for the system with triangle-shaped monomers): Assembly starts with forming a stable nucleus [dark blue shaded states in (a)], which requires a minimum number of $\sigma$ particles (nucleation size). The effective nucleation rate scales as $1/{\delta }_1^{\sigma -2}$, since $\sigma -2$ unstable states must be traversed to form the nucleus. Analogously, attachment processes are initiated by an effective reaction of the order of $\gamma$ (attachment order), after which a series of reactions of smaller (subleading) order may follow. As a particularity of triangular monomers, the first attachment reaction right after nucleation has attachment order $\gamma =4$, while all subsequent attachment reactions have a smaller attachment order $\gamma =3$.

Figure 1

Illustration of the model. (a) We consider the self-assembly of two-dimensional structures of volume (total number of constituents) $S$ from square-, triangle-, or hexagon-shaped building blocks and ask for the time the assembly processes take to reach substantial yields. The configurations shaded in dark blue, red, and yellow illustrate the first intermediate assembly states with enhanced stability (wherein each particle has at least two bonds). Typical assembly pathways consecutively pass through these (or equivalent) intermediate states. The dynamics can, thus, be effectively described by the three parameters $\sigma$, $\gamma$, and $\omega$, which are listed for the three cases and denote the nucleation size, attachment order, and growth exponent, respectively [see (c) and main text Secs. 3a and 4]. (b) Self-assembly dynamics is modeled as follows (illustrated for the system with square-shaped monomers): Two monomers dimerize at rate $\mu$ along any of their edges. Subsequently, monomers attach at rate $\nu$ to any free neighboring site until a square consisting of $S$ monomers with edge length $L=\sqrt{S}$ ($L=\sqrt{S/2}$ in the case of triangular monomers) is completed as illustrated in (a), assuming periodic boundary conditions between top and bottom and left and right. At the same time, particles can also detach from a given structure at rate ${\delta }_n\sim \exp [-n{E}_B/k_{B}T]$, with $n$ denoting the number of bonds between the particle and the structure and $E_B$ the binding energy per bond. (c) Coarse-grained effective dynamics (illustrated for the system with triangle-shaped monomers): Assembly starts with forming a stable nucleus [dark blue shaded states in (a)], which requires a minimum number of $\sigma$ particles (nucleation size). The effective nucleation rate scales as $1/{\delta }_1^{\sigma -2}$, since $\sigma -2$ unstable states must be traversed to form the nucleus. Analogously, attachment processes are initiated by an effective reaction of the order of $\gamma$ (attachment order), after which a series of reactions of smaller (subleading) order may follow. As a particularity of triangular monomers, the first attachment reaction right after nucleation has attachment order $\gamma =4$, while all subsequent attachment reactions have a smaller attachment order $\gamma =3$.

Figure 2

Typical cluster size distribution of a system with square-shaped monomers at a fixed, early time point ($T=0.1·T_{90}$) averaged over 100 independent simulation runs with target structure size $S=81$. Parameters: $E_B=10k_{B}T$ and $A=2.5\times {10}^{6}C\nu$. Error bars show the sample standard deviation for a few selected cluster sizes. Structure sizes that allow for stable configurations are significantly more frequent on average than those which do not allow for a stable configuration. A few of these stable configurations are illustrated explicitly. In an effective theory, one can exploit this feature and use a reduced state space that considers only the most likely connection pathways between these stable configurations; see Sec. 4.

Figure 2

Typical cluster size distribution of a system with square-shaped monomers at a fixed, early time point ($T=0.1·T_{90}$) averaged over 100 independent simulation runs with target structure size $S=81$. Parameters: $E_B=10k_{B}T$ and $A=2.5\times {10}^{6}C\nu$. Error bars show the sample standard deviation for a few selected cluster sizes. Structure sizes that allow for stable configurations are significantly more frequent on average than those which do not allow for a stable configuration. A few of these stable configurations are illustrated explicitly. In an effective theory, one can exploit this feature and use a reduced state space that considers only the most likely connection pathways between these stable configurations; see Sec. 4.

Figure 3

Final yield in the strong binding limit as a function of the detachment rate ${\tilde{\delta }}_1$ (${\delta }_1$ in units of $C\nu$) for different target structure sizes $S$ as indicated in the graph. Data points represent averages over ten independent runs of the stochastic simulation performed with square-shaped monomers and particle numbers $N=200S$. By increasing ${\tilde{\delta }}_1$, the final yield increases from 0 to a perfect value of 1. Larger target structure sizes require a larger value of ${\tilde{\delta }}_1$ to achieve a specified yield. Inset: Rescaling of the detachment rate ${\tilde{\delta }}_1$ by the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}(S)$ leads to a data collapse of the yield curves.

Figure 3

Final yield in the strong binding limit as a function of the detachment rate ${\tilde{\delta }}_1$ (${\delta }_1$ in units of $C\nu$) for different target structure sizes $S$ as indicated in the graph. Data points represent averages over ten independent runs of the stochastic simulation performed with square-shaped monomers and particle numbers $N=200S$. By increasing ${\tilde{\delta }}_1$, the final yield increases from 0 to a perfect value of 1. Larger target structure sizes require a larger value of ${\tilde{\delta }}_1$ to achieve a specified yield. Inset: Rescaling of the detachment rate ${\tilde{\delta }}_1$ by the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}(S)$ leads to a data collapse of the yield curves.

Figure 4

Assembly time and optimal detachment parameter in the strong binding limit. (a) The time $T_{90}$ required to achieve 90% yield in the stochastic simulations (markers) is plotted against the detachment rate ${\tilde{\delta }}_1$ (${\delta }_1$ in units of $C\nu$) for systems with triangle- (red), square- (blue), and hexagon-shaped (green) building blocks for a fixed target structure size $S=98$ (triangle-shaped building blocks) or $S=100$ (square- and hexagon-shaped building blocks). [Note that, in the case of triangle-shaped building blocks, there is no regular structure consisting of $S=100$ particles since regular structures consist of $2L^2$ (with an integer $L$) triangles; compare Fig. 1. Hence, we choose a structure size $S=98$ ($L=7$), which is as close as possible to the structure size $S=100$ for square and hexagonal monomers.] Simulations are performed with particle numbers $N=1000S$ (square- and hexagon-shaped building blocks) and $N=200S$ (triangle-shaped building blocks). Dash-dotted lines represent the prediction of the effective theory obtained by numerically integrating Eq. (4). For sufficiently large detachment rates, the assembly time scales proportionally to ${\tilde{\delta }}_1^{\sigma -2}$, while the assembly time diverges in the limit of small detachment rates. The value of ${\tilde{\delta }}_1$ at which the assembly time attains its minimum, $T_{90}^{\min }$, defines the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}$. The minimal assembly time $T_{90}^{\min }$ (b) and the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}$ (c) inferred from the stochastic simulations (markers and solid lines) and effective theory (dash-dotted lines) are plotted against the size $S$ of the target structure. They both exhibit approximate power-law dependencies for all three morphologies of the building blocks. Tables show the scaling exponents ${\theta }_{\mathrm{sim}}$ and ${\varphi }_{\mathrm{sim}}$ inferred from the last three data points of the stochastic simulation in comparison with their theoretical asymptotic values ${\theta }_{\mathrm{th}}$ and ${\varphi }_{\mathrm{th}}$ derived by the mathematical analysis; see Eqs. (14) and (17).

Figure 4

Assembly time and optimal detachment parameter in the strong binding limit. (a) The time $T_{90}$ required to achieve 90% yield in the stochastic simulations (markers) is plotted against the detachment rate ${\tilde{\delta }}_1$ (${\delta }_1$ in units of $C\nu$) for systems with triangle- (red), square- (blue), and hexagon-shaped (green) building blocks for a fixed target structure size $S=98$ (triangle-shaped building blocks) or $S=100$ (square- and hexagon-shaped building blocks). [Note that, in the case of triangle-shaped building blocks, there is no regular structure consisting of $S=100$ particles since regular structures consist of $2L^2$ (with an integer $L$) triangles; compare Fig. 1. Hence, we choose a structure size $S=98$ ($L=7$), which is as close as possible to the structure size $S=100$ for square and hexagonal monomers.] Simulations are performed with particle numbers $N=1000S$ (square- and hexagon-shaped building blocks) and $N=200S$ (triangle-shaped building blocks). Dash-dotted lines represent the prediction of the effective theory obtained by numerically integrating Eq. (4). For sufficiently large detachment rates, the assembly time scales proportionally to ${\tilde{\delta }}_1^{\sigma -2}$, while the assembly time diverges in the limit of small detachment rates. The value of ${\tilde{\delta }}_1$ at which the assembly time attains its minimum, $T_{90}^{\min }$, defines the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}$. The minimal assembly time $T_{90}^{\min }$ (b) and the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}$ (c) inferred from the stochastic simulations (markers and solid lines) and effective theory (dash-dotted lines) are plotted against the size $S$ of the target structure. They both exhibit approximate power-law dependencies for all three morphologies of the building blocks. Tables show the scaling exponents ${\theta }_{\mathrm{sim}}$ and ${\varphi }_{\mathrm{sim}}$ inferred from the last three data points of the stochastic simulation in comparison with their theoretical asymptotic values ${\theta }_{\mathrm{th}}$ and ${\varphi }_{\mathrm{th}}$ derived by the mathematical analysis; see Eqs. (14) and (17).

Figure 5

The nucleation process (illustrated for hexagon-shaped monomers) corresponds to a one-step Markov process with the nucleus as the absorbing state. The forward rates $\mu m$ and $2\nu m$ correspond to dimer formation and the subsequent assembly step, respectively, where the numerical prefactor in the second step accounts for the multiplicity of the process. Note that the forward rates are proportional to the monomer concentration $m$, considering the monomers as an unlimited reservoir. The backward rate equals $2{\delta }_1$, as either of the two monomers can detach. Hence, in the effective rate constant $\overline{\mu }$ for the nucleation process, Eq. (2), the combinatorial prefactors of the forward and backward reactions cancel out exactly. The same holds for the more complex nucleation processes for particles with square and triangular morphology.

Figure 5

The nucleation process (illustrated for hexagon-shaped monomers) corresponds to a one-step Markov process with the nucleus as the absorbing state. The forward rates $\mu m$ and $2\nu m$ correspond to dimer formation and the subsequent assembly step, respectively, where the numerical prefactor in the second step accounts for the multiplicity of the process. Note that the forward rates are proportional to the monomer concentration $m$, considering the monomers as an unlimited reservoir. The backward rate equals $2{\delta }_1$, as either of the two monomers can detach. Hence, in the effective rate constant $\overline{\mu }$ for the nucleation process, Eq. (2), the combinatorial prefactors of the forward and backward reactions cancel out exactly. The same holds for the more complex nucleation processes for particles with square and triangular morphology.

Figure 6

Illustration of the assembly steps for large clusters (illustrated for triangle-shaped monomers). Cluster growth after nucleation proceeds in two steps for assembly processes with attachment order $\gamma >1$ (here, $\gamma =3$): The formation of a seed of size $\gamma$ at one edge of the cluster (rate limiting slow process with an effective rate $p_s{q}_s\overline{\nu }{m}^{\gamma }$) is followed by a sequence of subleading fast processes that complete the edge very quickly (domino effect).

Figure 6

Illustration of the assembly steps for large clusters (illustrated for triangle-shaped monomers). Cluster growth after nucleation proceeds in two steps for assembly processes with attachment order $\gamma >1$ (here, $\gamma =3$): The formation of a seed of size $\gamma$ at one edge of the cluster (rate limiting slow process with an effective rate $p_s{q}_s\overline{\nu }{m}^{\gamma }$) is followed by a sequence of subleading fast processes that complete the edge very quickly (domino effect).

Figure 7

Time efficiency of the self-assembly of tubes. The minimal assembly time $T_{90}^{\min }$ for the self-assembly of two-dimensional tubes (cylinders) inferred from the stochastic simulations (markers and solid lines) and effective theory (dash-dotted lines) (cf. Appendix pp2-s6) is plotted against the size S of the target structure. The tubes have a fixed width (circumference) of six monomers and a variable height of $S/6$ monomers (with periodic boundary conditions in both dimensions). Simulations are performed with particle numbers $N=500S$. The minimal assembly time is determined by averaging the assembly times over ten independent runs for each different value of ${\delta }_1$ and subsequently taking the minimum of the averages. The table shows the time complexity exponent ${\theta }_{\mathrm{sim}}$ inferred from the last three data points of the stochastic simulation in comparison with the theoretical value ${\theta }_{\mathrm{th}}$ according to Eq. (17) with growth exponent $\omega =0$. Predictions of the effective model agree excellently with the simulation data. The fact that the exponents are very different from those in Fig. 4 shows that also the morphology of the target structure crucially influences self-assembly efficiency.

Figure 7

Time efficiency of the self-assembly of tubes. The minimal assembly time $T_{90}^{\min }$ for the self-assembly of two-dimensional tubes (cylinders) inferred from the stochastic simulations (markers and solid lines) and effective theory (dash-dotted lines) (cf. Appendix pp2-s6) is plotted against the size S of the target structure. The tubes have a fixed width (circumference) of six monomers and a variable height of $S/6$ monomers (with periodic boundary conditions in both dimensions). Simulations are performed with particle numbers $N=500S$. The minimal assembly time is determined by averaging the assembly times over ten independent runs for each different value of ${\delta }_1$ and subsequently taking the minimum of the averages. The table shows the time complexity exponent ${\theta }_{\mathrm{sim}}$ inferred from the last three data points of the stochastic simulation in comparison with the theoretical value ${\theta }_{\mathrm{th}}$ according to Eq. (17) with growth exponent $\omega =0$. Predictions of the effective model agree excellently with the simulation data. The fact that the exponents are very different from those in Fig. 4 shows that also the morphology of the target structure crucially influences self-assembly efficiency.

Figure 8

Examples for morphologies that self-assemble efficiently in the strong binding limit. Squares with six instead of four binding sites form structures topologically equivalent to those formed by hexagons (cf. Fig. 1) (i.e., the squares in the finished structure can be smoothly deformed into hexagons without breaking any bonds). The same color and number indicate matching bonds. Hence, self-assembly of the squares is described by the same kinetic parameters ($\sigma =3$, $\gamma =1$, and $\omega \approx 1/2$) as for hexagons and exhibits the same minimal time complexity exponent $\theta \approx 1/2$. (b) Using two or more different types of constituents (here, octagons and squares) offers additional possibilities for designing efficient assembly schemes. For the example illustrated, again $\sigma =3$, $\gamma =1$, and $\omega \approx 1/2$ and, hence, $\theta \approx 1/2$ as for hexagons.

Figure 8

Examples for morphologies that self-assemble efficiently in the strong binding limit. Squares with six instead of four binding sites form structures topologically equivalent to those formed by hexagons (cf. Fig. 1) (i.e., the squares in the finished structure can be smoothly deformed into hexagons without breaking any bonds). The same color and number indicate matching bonds. Hence, self-assembly of the squares is described by the same kinetic parameters ($\sigma =3$, $\gamma =1$, and $\omega \approx 1/2$) as for hexagons and exhibits the same minimal time complexity exponent $\theta \approx 1/2$. (b) Using two or more different types of constituents (here, octagons and squares) offers additional possibilities for designing efficient assembly schemes. For the example illustrated, again $\sigma =3$, $\gamma =1$, and $\omega \approx 1/2$ and, hence, $\theta \approx 1/2$ as for hexagons.

Figure 9

Hierarchical self-assembly as a way to design optimized assembly schemes. Hierarchically organizing the assembly process can increase self-assembly efficiency significantly by optimizing the morphology of intermediate assembly products. (a) To induce triangle-shaped monomers to assemble hierarchically, bonds between edges indicated in brown are assumed to be more stable than bonds between black edges. As a result of this variation of the binding strengths, monomers first form hexamers (hexagons) or dimers (rhomboids), respectively, which subsequently assemble into the structures more efficiently than the original triangles; cf. Fig. 4. (b) The assembly time $T_{90}$ is plotted against the detachment rate ${\tilde{\delta }}_1$ for the system with triangle-shaped monomers in comparison with the two hierarchical assembly schemes illustrated in (a) for a target structure of size $S=1200$. Simulations are performed with the effective model (see Appendix pp2), assuming a constant ratio $r_h={10}^{-4}$ between the detachment rates of strong and weak bonds. Since, according to our findings, the self-assembly of particles with a hexagonal morphology is remarkably efficient, the assembly time can, thus, be reduced by 2–3 orders of magnitude over a broad range of the control parameter ${\tilde{\delta }}_1$.

Figure 9

Hierarchical self-assembly as a way to design optimized assembly schemes. Hierarchically organizing the assembly process can increase self-assembly efficiency significantly by optimizing the morphology of intermediate assembly products. (a) To induce triangle-shaped monomers to assemble hierarchically, bonds between edges indicated in brown are assumed to be more stable than bonds between black edges. As a result of this variation of the binding strengths, monomers first form hexamers (hexagons) or dimers (rhomboids), respectively, which subsequently assemble into the structures more efficiently than the original triangles; cf. Fig. 4. (b) The assembly time $T_{90}$ is plotted against the detachment rate ${\tilde{\delta }}_1$ for the system with triangle-shaped monomers in comparison with the two hierarchical assembly schemes illustrated in (a) for a target structure of size $S=1200$. Simulations are performed with the effective model (see Appendix pp2), assuming a constant ratio $r_h={10}^{-4}$ between the detachment rates of strong and weak bonds. Since, according to our findings, the self-assembly of particles with a hexagonal morphology is remarkably efficient, the assembly time can, thus, be reduced by 2–3 orders of magnitude over a broad range of the control parameter ${\tilde{\delta }}_1$.

Figure 10

Illustration of the nucleation process for triangle-shaped monomers. Starting from a monomer, the nucleus is formed by subsequent attachment of monomers with growth rates $g_i\sim \nu m$ indicated in the graph. The detachment of monomers counteracts this with the shrinkage (detachment) rate $d_i=2{\delta }_1$. The prefactors account for the degeneracy of attachment or detachment reactions leading to equivalent configurations. In the limiting case of large detachment rates, the effective nucleation rate $\overline{\mu }$ is given by the product of the forward rates divided by the product of the reverse rates.

Figure 10

Illustration of the nucleation process for triangle-shaped monomers. Starting from a monomer, the nucleus is formed by subsequent attachment of monomers with growth rates $g_i\sim \nu m$ indicated in the graph. The detachment of monomers counteracts this with the shrinkage (detachment) rate $d_i=2{\delta }_1$. The prefactors account for the degeneracy of attachment or detachment reactions leading to equivalent configurations. In the limiting case of large detachment rates, the effective nucleation rate $\overline{\mu }$ is given by the product of the forward rates divided by the product of the reverse rates.

Figure 11

Derivation of the multiplicity factor $q_s$ for seed formation for (a) nuclei and (b) larger clusters with triangle-shaped monomers. The diagram depicts all equivalent assembly pathways by which a seed can be completed after the first seed-forming monomer has attached. The rate for all forward steps in the diagram is $\nu m$, while the rates for the backward steps are given by ${\delta }_1$ or $2{\delta }_1$, depending on whether there are one or two monomers with a single bond. The total weight of each path (indicated above the rightmost arrows) is calculated by multiplying all backward rates along the respective path, i.e., by applying Eq. (b5) to the path. The multiplicity factor $q_s$ corresponds to the sum of the weights of all contributing paths. Note that $q_s$ has an additional factor of 2, because one must additionally account for the analogous pathways in which the seed forms to the left-hand side of the first monomer instead of to the right.

Figure 11

Derivation of the multiplicity factor $q_s$ for seed formation for (a) nuclei and (b) larger clusters with triangle-shaped monomers. The diagram depicts all equivalent assembly pathways by which a seed can be completed after the first seed-forming monomer has attached. The rate for all forward steps in the diagram is $\nu m$, while the rates for the backward steps are given by ${\delta }_1$ or $2{\delta }_1$, depending on whether there are one or two monomers with a single bond. The total weight of each path (indicated above the rightmost arrows) is calculated by multiplying all backward rates along the respective path, i.e., by applying Eq. (b5) to the path. The multiplicity factor $q_s$ corresponds to the sum of the weights of all contributing paths. Note that $q_s$ has an additional factor of 2, because one must additionally account for the analogous pathways in which the seed forms to the left-hand side of the first monomer instead of to the right.

Figure 12

Effective assembly process in the strong binding limit, illustrated for square-shaped monomers. Top: typical assembly path leading from a stable nucleus of size $\sigma =4$ to a structure of size $s=16$, with the corresponding transition rates between the states, indicated in the graph (assuming ${\delta }_n=0$ for $n\le 2$). The prefactor $p_s$ denotes the average perimeter (number of binding sites for the seed) of a cluster of size $s$. The effective theory is derived in two steps illustrated by the following. Middle: First, we describe transitions passing through unstable states as effective few-particle reactions of the order of $\gamma$ (here, for square-shaped monomers, $\gamma =2$) with rate $r_{s\to s+\gamma }^{\mathrm{seed}}$. This is possible, since the backward rate ${\delta }_1$ is large compared to the forward rates (cf. Appendix pp2). This way, we “integrate out” the fast detachment rate ${\delta }_1$ from the process. Bottom: In the second step, we introduce one-step forward rates $g_s$, which are continuous functions of the structure size $s$. These average one-step rates are calculated by dividing the number of particles added per row by the total time it takes to add the entire row. In other words, this corresponds to taking a moving harmonic average of the rates of the leading-order reactions and the subsequent subleading-order reactions as indicated by the square brackets. Since the leading-order reactions are significantly slower than the subleading-order reactions, the latter can be assumed as infinitely fast. In this way, we derive a one-step Poisson process with transition rates that vary continuously as a function of the cluster size. This enables a convenient mathematical analysis and allows us to describe systems with different particle morphologies on equal footing.

Figure 12

Effective assembly process in the strong binding limit, illustrated for square-shaped monomers. Top: typical assembly path leading from a stable nucleus of size $\sigma =4$ to a structure of size $s=16$, with the corresponding transition rates between the states, indicated in the graph (assuming ${\delta }_n=0$ for $n\le 2$). The prefactor $p_s$ denotes the average perimeter (number of binding sites for the seed) of a cluster of size $s$. The effective theory is derived in two steps illustrated by the following. Middle: First, we describe transitions passing through unstable states as effective few-particle reactions of the order of $\gamma$ (here, for square-shaped monomers, $\gamma =2$) with rate $r_{s\to s+\gamma }^{\mathrm{seed}}$. This is possible, since the backward rate ${\delta }_1$ is large compared to the forward rates (cf. Appendix pp2). This way, we “integrate out” the fast detachment rate ${\delta }_1$ from the process. Bottom: In the second step, we introduce one-step forward rates $g_s$, which are continuous functions of the structure size $s$. These average one-step rates are calculated by dividing the number of particles added per row by the total time it takes to add the entire row. In other words, this corresponds to taking a moving harmonic average of the rates of the leading-order reactions and the subsequent subleading-order reactions as indicated by the square brackets. Since the leading-order reactions are significantly slower than the subleading-order reactions, the latter can be assumed as infinitely fast. In this way, we derive a one-step Poisson process with transition rates that vary continuously as a function of the cluster size. This enables a convenient mathematical analysis and allows us to describe systems with different particle morphologies on equal footing.

Figure 13

Regular hexagonal cluster consisting of $s=54$ triangular building blocks to illustrate the relation between $e_s$ (number of binding sites per edge), $l_s$ (number of particles forming an edge), and the cluster size $s$. The cluster size $s$ equates to $s=2{\sum }_{k=0}^{e_s-1}(l_s+2k)$, which can be calculated to yield the relation shown in the figure. In the limit of large $s$, one thus finds $s\approx \frac{3}{2}{l}_s^2$.

Figure 13

Regular hexagonal cluster consisting of $s=54$ triangular building blocks to illustrate the relation between $e_s$ (number of binding sites per edge), $l_s$ (number of particles forming an edge), and the cluster size $s$. The cluster size $s$ equates to $s=2{\sum }_{k=0}^{e_s-1}(l_s+2k)$, which can be calculated to yield the relation shown in the figure. In the limit of large $s$, one thus finds $s\approx \frac{3}{2}{l}_s^2$.

Figure 14

Assembly time and optimal detachment parameter for three-dimensional structures. (a) The assembly time $T_{90}$ for the self-assembly of cube-shaped monomers into three-dimensional cubic structures is plotted against the detachment rate ${\delta }_1$. Markers show the results of stochastic simulations, while dash-dotted lines show the prediction of the effective model for three different sizes of the target structure. Stochastic simulations are performed with particle numbers $N=500S$ and averaged over ten independent runs. The effective model is integrated with parameters $\sigma =4$, $\gamma =2$, $\omega =4/3$, $a=15$, and $\mu =2\nu$; see Appendix pp4-s2. (b) The minimal assembly time $T_{90}^{\min }$ and the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}$ inferred from the stochastic simulations (markers and solid lines) and effective theory (dash-dotted lines) are plotted against the size $S$ of the target structure. The tables show their respective scaling exponents inferred from the last three data points of the stochastic simulation in comparison with their theoretical values according to Eq. (d17).

Figure 14

Assembly time and optimal detachment parameter for three-dimensional structures. (a) The assembly time $T_{90}$ for the self-assembly of cube-shaped monomers into three-dimensional cubic structures is plotted against the detachment rate ${\delta }_1$. Markers show the results of stochastic simulations, while dash-dotted lines show the prediction of the effective model for three different sizes of the target structure. Stochastic simulations are performed with particle numbers $N=500S$ and averaged over ten independent runs. The effective model is integrated with parameters $\sigma =4$, $\gamma =2$, $\omega =4/3$, $a=15$, and $\mu =2\nu$; see Appendix pp4-s2. (b) The minimal assembly time $T_{90}^{\min }$ and the optimal detachment rate ${\tilde{\delta }}_1^{\mathrm{opt}}$ inferred from the stochastic simulations (markers and solid lines) and effective theory (dash-dotted lines) are plotted against the size $S$ of the target structure. The tables show their respective scaling exponents inferred from the last three data points of the stochastic simulation in comparison with their theoretical values according to Eq. (d17).

Figure 15

Beyond ideal self-assembly: the role of cluster-cluster interactions. (a) The ideality index $I$, which estimates the number of interactions of an oligomer with other oligomers until time $T_{90}$ [see definition Eq. (e1)], is plotted against the detachment rate ${\tilde{\delta }}_1$ for the assembly processes with three different particle morphologies ($S=100$). The index is calculated numerically by integrating the effective model, Eq. (4). Values of $I$ larger than 1 (dashed line) indicate that cluster interactions may be important for the assembly dynamics. Thus, a strong dependence of the potential role of cluster interactions on particle morphology is revealed. (b) Assembly time $T_{90}$ with (drawn line) and without (dotted line) cluster interactions as a function of the detachment rate ${\tilde{\delta }}_1$. Assembly times are obtained by integrating the effective theory that is extended by additional terms accounting for all possible cluster reactions $c_s+c_p\stackrel{\nu }{⟶}{c}_{s+p}$ subject to the constraint $s+p\le S$ (see Appendix pp2 for details). Since, in an actual system, the rate of reactions between clusters would likely be smaller, this must be considered an upper estimate of the effect expected by cluster interactions. The minimal assembly time is only slightly reduced as a consequence of cluster interactions, and—as in the ideal case—the assembly time increases like $T_{90}\sim {\tilde{\delta }}_1^{\sigma -2}$ for large ${\tilde{\delta }}_1$.

Figure 15

Beyond ideal self-assembly: the role of cluster-cluster interactions. (a) The ideality index $I$, which estimates the number of interactions of an oligomer with other oligomers until time $T_{90}$ [see definition Eq. (e1)], is plotted against the detachment rate ${\tilde{\delta }}_1$ for the assembly processes with three different particle morphologies ($S=100$). The index is calculated numerically by integrating the effective model, Eq. (4). Values of $I$ larger than 1 (dashed line) indicate that cluster interactions may be important for the assembly dynamics. Thus, a strong dependence of the potential role of cluster interactions on particle morphology is revealed. (b) Assembly time $T_{90}$ with (drawn line) and without (dotted line) cluster interactions as a function of the detachment rate ${\tilde{\delta }}_1$. Assembly times are obtained by integrating the effective theory that is extended by additional terms accounting for all possible cluster reactions $c_s+c_p\stackrel{\nu }{⟶}{c}_{s+p}$ subject to the constraint $s+p\le S$ (see Appendix pp2 for details). Since, in an actual system, the rate of reactions between clusters would likely be smaller, this must be considered an upper estimate of the effect expected by cluster interactions. The minimal assembly time is only slightly reduced as a consequence of cluster interactions, and—as in the ideal case—the assembly time increases like $T_{90}\sim {\tilde{\delta }}_1^{\sigma -2}$ for large ${\tilde{\delta }}_1$.

Figure 16

Assembly time $T_{50}$ (solid) in comparison with assembly time $T_{90}$ (faded). (a) The assembly times $T_{50}$ (solid) and $T_{90}$ (faded) required to achieve 50% or 90% yield, respectively, in the strong binding limit are plotted against the detachment rate ${\delta }_1$ (in units of $C\nu$) for systems with triangle- (red), square- (blue), and hexagon-shaped (green) building blocks for a fixed target structure size $S=98$ (triangle-shaped monomers) or $S=100$ (square and hexagonal monomers). Markers represent averages of ten independent runs of the stochastic simulation [performed with $N=500S$ and $N=300S$ (triangle-shaped monomers)]. The dash-dotted colored lines represent the prediction of the effective theory obtained by numerically integrating Eq. (4). Black dashed lines show the analytic result for the assembly times given by Eqs. (31) and (32) in the slow nucleation regime (ranges for ${\tilde{\delta }}_1$ are chosen manually). (b) The minimal assembly time $T_{50}^{\min }$ (solid) and $T_{90}^{\min }$ (faded) inferred from the stochastic simulation (markers and solid lines) and effective theory (dash-dotted lines) are plotted against the size $S$ of the target structure. Although the assembly times $T_{90}$ and $T_{50}$ may differ significantly, they exhibit approximately the same scaling in dependence on the structure size $S$. The scaling exponents ${\theta }_{\mathrm{sim}}$ inferred from the last three data points of the stochastic simulation for $T_{50}$ [and their theoretical asymptotic values ${\theta }_{\mathrm{th}}$; cf. Eq. (17)] are summarized in the table [compare with the analogous exponents ${\theta }_{\mathrm{sim}}$ for $T_{90}$ exhibited in the table in Fig. 4].

Figure 16

Assembly time $T_{50}$ (solid) in comparison with assembly time $T_{90}$ (faded). (a) The assembly times $T_{50}$ (solid) and $T_{90}$ (faded) required to achieve 50% or 90% yield, respectively, in the strong binding limit are plotted against the detachment rate ${\delta }_1$ (in units of $C\nu$) for systems with triangle- (red), square- (blue), and hexagon-shaped (green) building blocks for a fixed target structure size $S=98$ (triangle-shaped monomers) or $S=100$ (square and hexagonal monomers). Markers represent averages of ten independent runs of the stochastic simulation [performed with $N=500S$ and $N=300S$ (triangle-shaped monomers)]. The dash-dotted colored lines represent the prediction of the effective theory obtained by numerically integrating Eq. (4). Black dashed lines show the analytic result for the assembly times given by Eqs. (31) and (32) in the slow nucleation regime (ranges for ${\tilde{\delta }}_1$ are chosen manually). (b) The minimal assembly time $T_{50}^{\min }$ (solid) and $T_{90}^{\min }$ (faded) inferred from the stochastic simulation (markers and solid lines) and effective theory (dash-dotted lines) are plotted against the size $S$ of the target structure. Although the assembly times $T_{90}$ and $T_{50}$ may differ significantly, they exhibit approximately the same scaling in dependence on the structure size $S$. The scaling exponents ${\theta }_{\mathrm{sim}}$ inferred from the last three data points of the stochastic simulation for $T_{50}$ [and their theoretical asymptotic values ${\theta }_{\mathrm{th}}$; cf. Eq. (17)] are summarized in the table [compare with the analogous exponents ${\theta }_{\mathrm{sim}}$ for $T_{90}$ exhibited in the table in Fig. 4].

Figure 17

Yield and assembly time in seeded systems (according to the effective model). (a) The maximal yield of target structures of size $S=100$ in the effective model in the limit of large ${\tilde{\delta }}_1$ (here, ${\tilde{\delta }}_1={10}^{12}$) is plotted against the initial concentration $\mathrm{\Sigma }$ of nuclei (seeds). Parameters of the effective model are chosen to simulate the self-assembly of square-shaped monomers ($\sigma =4$, $\gamma =2$, $\omega =1$, and $a=5.3$), and the initial monomer concentration is $m(0)=C-\sigma \mathrm{\Sigma }$. (b) Assembly time $T_{90}$ as a function of ${\tilde{\delta }}_1$ for seeded systems with initial seed concentration $\mathrm{\Sigma }=C/S$ (drawn lines) and corresponding unseeded systems (dash-dotted lines, faded). (c) Minimal assembly time $T_{90}^{\min }$ as a function of $S$ for seeded systems with initial seed concentration $\mathrm{\Sigma }=C/S$ (drawn lines) and corresponding unseeded systems (dash-dotted lines, faded). The black dashed line shows the assembly time for a putative system of triangular monomers in which the first assembly step after nucleation has attachment order ${\gamma }^{\prime }=3$ instead of ${\gamma }^{\prime }=4$. The table shows the time complexity exponents ${\theta }_{\mathrm{sim}}$ measured in the range of target structure sizes $400\le S\le 1000$ and their corresponding theoretical values ${\theta }_{\mathrm{th}}$.

Figure 17

Yield and assembly time in seeded systems (according to the effective model). (a) The maximal yield of target structures of size $S=100$ in the effective model in the limit of large ${\tilde{\delta }}_1$ (here, ${\tilde{\delta }}_1={10}^{12}$) is plotted against the initial concentration $\mathrm{\Sigma }$ of nuclei (seeds). Parameters of the effective model are chosen to simulate the self-assembly of square-shaped monomers ($\sigma =4$, $\gamma =2$, $\omega =1$, and $a=5.3$), and the initial monomer concentration is $m(0)=C-\sigma \mathrm{\Sigma }$. (b) Assembly time $T_{90}$ as a function of ${\tilde{\delta }}_1$ for seeded systems with initial seed concentration $\mathrm{\Sigma }=C/S$ (drawn lines) and corresponding unseeded systems (dash-dotted lines, faded). (c) Minimal assembly time $T_{90}^{\min }$ as a function of $S$ for seeded systems with initial seed concentration $\mathrm{\Sigma }=C/S$ (drawn lines) and corresponding unseeded systems (dash-dotted lines, faded). The black dashed line shows the assembly time for a putative system of triangular monomers in which the first assembly step after nucleation has attachment order ${\gamma }^{\prime }=3$ instead of ${\gamma }^{\prime }=4$. The table shows the time complexity exponents ${\theta }_{\mathrm{sim}}$ measured in the range of target structure sizes $400\le S\le 1000$ and their corresponding theoretical values ${\theta }_{\mathrm{th}}$.

Figure 18

Final cluster size distribution. The final cluster size distributions for (a) hexagon-shaped ($\omega \approx 1/2>1$), (b) square-shaped ($\omega =1$), and (c) 3D cube-shaped ($\omega =4/3$) monomers are shown for the respective values of the detachment rate ${\delta }_1$ indicated in each plot. The target structure size in the simulation is chosen very large [$S=3025$ (a),(b) and $S=8000$ (c)] so that the yield is zero in each case, thereby effectively simulating unlimited cluster growth. The initial number of monomers is $N=3\times {10}^5$, and the distribution is averaged over 40 independent simulation runs (blue circles). Subsequently, moving averages are taken over windows of $k=10$ (red circles) and $k=50$ (green circles) cluster sizes. The averaged cluster size distribution decreases exponentially for large cluster sizes for square- and cube-shaped monomers (i.e., for $\omega \ge 1$). Black lines show the final cluster size distribution obtained by integrating the effective model.

Figure 18

Final cluster size distribution. The final cluster size distributions for (a) hexagon-shaped ($\omega \approx 1/2>1$), (b) square-shaped ($\omega =1$), and (c) 3D cube-shaped ($\omega =4/3$) monomers are shown for the respective values of the detachment rate ${\delta }_1$ indicated in each plot. The target structure size in the simulation is chosen very large [$S=3025$ (a),(b) and $S=8000$ (c)] so that the yield is zero in each case, thereby effectively simulating unlimited cluster growth. The initial number of monomers is $N=3\times {10}^5$, and the distribution is averaged over 40 independent simulation runs (blue circles). Subsequently, moving averages are taken over windows of $k=10$ (red circles) and $k=50$ (green circles) cluster sizes. The averaged cluster size distribution decreases exponentially for large cluster sizes for square- and cube-shaped monomers (i.e., for $\omega \ge 1$). Black lines show the final cluster size distribution obtained by integrating the effective model.

Figure 19

Heterogeneous systems. Simulating the self-assembly of heterogeneous structures (information-rich structures), we assume that each species (labeled by an index $1\dots S$) is unique and binds only with the respective neighboring species, as shown. Thus, each species appears only once in a finished structure and at a specified position. The underlying kinetics for self-assembly are the same as in the homogeneous case, but the different species can have distinct binding and detachment rates or varying concentrations. In Appendix pp10, we investigate how self-assembly behaves for various heterogeneous systems.

Figure 19

Heterogeneous systems. Simulating the self-assembly of heterogeneous structures (information-rich structures), we assume that each species (labeled by an index $1\dots S$) is unique and binds only with the respective neighboring species, as shown. Thus, each species appears only once in a finished structure and at a specified position. The underlying kinetics for self-assembly are the same as in the homogeneous case, but the different species can have distinct binding and detachment rates or varying concentrations. In Appendix pp10, we investigate how self-assembly behaves for various heterogeneous systems.

Figure 20

Robustness of the scaling behavior. The minimal time required to achieve 90% yield ($T_{90}^{\min }$) is plotted in dependence of the target structure size $S$ for different variants of the model. Diamond and triangle markers show $T_{90}^{\min }$ for heterogeneous systems with random heterogeneous binding rates drawn from a (truncated) Gaussian with coefficient of variation 50% (cf. Appendix pp10-s2) or structures with explicit boundaries (cf. Appendix pp10-s3), respectively. Squares and circles show $T_{90}^{\min }$ for the heterogeneous system with identical rates and periodic boundaries [equivalent to the homogeneous system; cf. Fig. 4] as well as for a homogeneous system under annealing [i.e., with a time-dependent detachment rate ${\delta }_1(t)\sim m(t)$] (cf. Appendix pp8). Data points are obtained by averaging 10–20 independent runs of the stochastic simulation performed with $N=500$ particles per species (heterogeneous case) or $N=500S$ particles (homogeneous case). While the assembly time varies for the different model variants, the measured time complexity exponents are largely invariant.

Figure 20

Robustness of the scaling behavior. The minimal time required to achieve 90% yield ($T_{90}^{\min }$) is plotted in dependence of the target structure size $S$ for different variants of the model. Diamond and triangle markers show $T_{90}^{\min }$ for heterogeneous systems with random heterogeneous binding rates drawn from a (truncated) Gaussian with coefficient of variation 50% (cf. Appendix pp10-s2) or structures with explicit boundaries (cf. Appendix pp10-s3), respectively. Squares and circles show $T_{90}^{\min }$ for the heterogeneous system with identical rates and periodic boundaries [equivalent to the homogeneous system; cf. Fig. 4] as well as for a homogeneous system under annealing [i.e., with a time-dependent detachment rate ${\delta }_1(t)\sim m(t)$] (cf. Appendix pp8). Data points are obtained by averaging 10–20 independent runs of the stochastic simulation performed with $N=500$ particles per species (heterogeneous case) or $N=500S$ particles (homogeneous case). While the assembly time varies for the different model variants, the measured time complexity exponents are largely invariant.

Figure 21

Heterogeneous concentrations and heterogeneous detachment rates. The dark blue line with crosses shows the minimal assembly time of a heterogeneous system in which species 1, 2, $L+1$, and $L+2$ (cf. Fig. 19) are provided in sixfold higher concentration compared to the other species. The bright blue line with diamond markers shows the minimal assembly time when the detachment rates for bonds between species 1, 2, $L+1$, and $L+2$ are set to zero. In both cases, data points are obtained by averaging ten independent runs of the stochastic simulation performed with $N=500$ particles per species and a dimerization barrier $\tilde{\mu }≔\mu /\nu =0.1$ to increase the efficiency of the simulations (cf. Appendix pp4-s1) (note that the $y$ axis is scaled in units of $\tilde{\mu }$ accordingly). For comparison, the blue dash-dotted line shows $T_{90}^{\min }$ for the homogeneous system under seeding determined with the effective model (cf. Appendix pp7), and the purple line with square markers shows $T_{90}^{\min }$ for the heterogeneous system with identical rates and concentrations [equivalent to the homogeneous system; cf. Fig. 4].

Figure 21

Heterogeneous concentrations and heterogeneous detachment rates. The dark blue line with crosses shows the minimal assembly time of a heterogeneous system in which species 1, 2, $L+1$, and $L+2$ (cf. Fig. 19) are provided in sixfold higher concentration compared to the other species. The bright blue line with diamond markers shows the minimal assembly time when the detachment rates for bonds between species 1, 2, $L+1$, and $L+2$ are set to zero. In both cases, data points are obtained by averaging ten independent runs of the stochastic simulation performed with $N=500$ particles per species and a dimerization barrier $\tilde{\mu }≔\mu /\nu =0.1$ to increase the efficiency of the simulations (cf. Appendix pp4-s1) (note that the $y$ axis is scaled in units of $\tilde{\mu }$ accordingly). For comparison, the blue dash-dotted line shows $T_{90}^{\min }$ for the homogeneous system under seeding determined with the effective model (cf. Appendix pp7), and the purple line with square markers shows $T_{90}^{\min }$ for the heterogeneous system with identical rates and concentrations [equivalent to the homogeneous system; cf. Fig. 4].

Figure 22

Improving heterogeneous self-assembly: hierarchical self-assembly and heterogeneous dimerization rates. The blue dash-dotted line shows the minimal assembly time obtained with the effective model for the hierarchical self-assembly scenario in which square-shaped monomers first form rectangle-shaped dimers (${\delta }_1^{\text{strong}}=0$), which then assemble on gap as shown in Fig. 8. Notice that this hierarchical scenario can be realized only with heterogeneous building blocks. The dark blue line with square markers shows the minimal assembly time of a heterogeneous system in which only species 1 and 2 can dimerize with rate ${\mu }_{12}=\nu$ and the dimerization rates of all other pairs of species as well as the detachment rates are zero. Simulations are performed with $N=1000$ particles per species and averaged over 10 independent runs. The purple line with square markers shows the assembly time in a heterogeneous system with identical rates for comparison. Furthermore, the green line with star markers and the green dash-dotted line show the assembly time of a homogeneous system with hexagonal monomers (stochastic simulation and effective model, respectively) copied from Fig. 4 for comparison. With hierarchical self-assembly or heterogeneous detachment rates, the required assembly time can be reduced drastically.

Figure 22

Improving heterogeneous self-assembly: hierarchical self-assembly and heterogeneous dimerization rates. The blue dash-dotted line shows the minimal assembly time obtained with the effective model for the hierarchical self-assembly scenario in which square-shaped monomers first form rectangle-shaped dimers (${\delta }_1^{\text{strong}}=0$), which then assemble on gap as shown in Fig. 8. Notice that this hierarchical scenario can be realized only with heterogeneous building blocks. The dark blue line with square markers shows the minimal assembly time of a heterogeneous system in which only species 1 and 2 can dimerize with rate ${\mu }_{12}=\nu$ and the dimerization rates of all other pairs of species as well as the detachment rates are zero. Simulations are performed with $N=1000$ particles per species and averaged over 10 independent runs. The purple line with square markers shows the assembly time in a heterogeneous system with identical rates for comparison. Furthermore, the green line with star markers and the green dash-dotted line show the assembly time of a homogeneous system with hexagonal monomers (stochastic simulation and effective model, respectively) copied from Fig. 4 for comparison. With hierarchical self-assembly or heterogeneous detachment rates, the required assembly time can be reduced drastically.