Entropy directs the self-assembly of supramolecular palladium coordination macrocycles and cages

The self-assembly of palladium-based cages is frequently rationalized via the cumulative enthalpy (ΔH) of bonds between coordination nodes (M, i.e., Pd) and ligand (L) components. This focus on enthalpic rationale limits the complete understanding of the Gibbs free energy (ΔG) for self-assembly, as entropic (ΔS) contributions are overlooked. Here, we present a study of the M2linL3 intermediate species (M = dinitrato(N,N,N′,N′-tetramethylethylenediamine)palladium(ii), linL = 4,4′-bipyridine), formed during the synthesis of triangle-shaped (M3linL3) and square-shaped (M4linL4) coordination macrocycles. Thermochemical analyses by variable temperature (VT) 1H-NMR revealed that the M2linL3 intermediate exhibited an unfavorable (relative) ΔS compared to M3linL3 (triangle, ΔTΔS = +5.22 kcal mol−1) or M4linL4 (square, ΔTΔS = +2.37 kcal mol−1) macrocycles. Further analysis of these constructs with molecular dynamics (MD) identified that the self-assembly process is driven by ΔG losses facilitated by increases in solvation entropy (ΔSsolv, i.e., depletion of solvent accessible surface area) that drives the self-assembly from “open” intermediates toward “closed” macrocyclic products. Expansion of our computational approach to the analysis of self-assembly in PdnbenL2n cages (benL = 4,4'-(5-ethoxy-1,3-phenylene)dipyridine), demonstrated that ΔSsolv contributions drive the self-assembly of both thermodynamic cage products (i.e., Pd12benL24) and kinetically-trapped intermediates (i.e., Pd8cL16).

The self-assembly of equimolar amounts of dinitrato(N,N,N 0 ,N 0 -tetramethylethylenediamine)palladium(II) (M) and 4,4 0 -bipyridine ( lin L) affords a mixture of triangular and square macrocycles in equilibrium (Scheme 1). 1,2 Previous reports have leveraged this experimentally accessible equilibrium to measure the relative DS and DH of triangular and square complexes. 5 These studies found that DS favors the assembly featuring fewer components (i.e., triangles) while DH favors the geometric matching between the squareplanar metal center and the ligand geometry (i.e., squares). 1,5 The synthesis of (and conversion between) macrocyclic assemblies proceeds via coordinatively unsaturated oligomeric intermediates (Scheme 1, purple).
Interestingly, similar stable oligomer intermediates have been found in the synthesis of polygonal organometallic macrocyclic assemblies. [1][2][3][4][5] The equilibrium between these stable oligomeric intermediates and macrocyclic products may be leveraged to quantify DH and DS contributions to selfassembly using the literature described NMR-based approach. 5 Realization of the origin and effect of these thermodynamic contributions enables rational improvement of the self-assembly of highly-ordered constructs used broadly in supramolecular chemistry, including coordination cages, metal-organic frameworks, and dynamic-covalent based constructs. 52,[59][60][61][62] In this report, we demonstrate that the self-assembly of an equimolar mixture of M and lin L in deuterated dimethyl sulfoxide (DMSO) affords a mixture of triangular (M 3 lin L 3 ) and square (M 4 lin L 4 ), and oligomeric (M n lin L n+1 ) assemblies as depicted in Scheme 1. 1, 6 We employed variable temperature 1 H-NMR (VT-NMR) to determine the DH and DS of both oligomeric and macrocyclic assemblies, providing unprecedented thermodynamic insights into the self-assembly process. Importantly, this thermodynamic data-enabled validation of a molecular dynamics (MD) 52 based approach to distinguish respective DS contributions arising from the assembly structure (DS struct , eqn S1 †) 53 and its solvation (DS solv , eqn S2 †). 54 These individual entropic contributions, alongside calculation of DH (eqn S3 †), ultimately provide an accurate DG for self-assembly. We applied our MD-based approach to the study of coordination cages based on 4,4'-(5-ethoxy-1,3-phenylene)dipyridine as a bent ditopic ligand ( ben L) and free palladium(II) ions (Pd 2+ ), which have been reported in the literature (Scheme 2). [8][9][10][11][12][13] Thermodynamic estimates derived from MD simulations reveal a DS solv -driven, self-assembly process for macrocycles and cages reminiscent of biopolymer folding. 51 The generalization of our MD-based approach may distinguish between kinetically accessible thermodynamic products (i.e., Pd 12 ben L 24 ) and undesirable kinetically-trapped intermediate assemblies (e.g. Pd 8 ben L 16 ). 55 These computational and experimental studies demonstrate that DS drives the self-assembly of supramolecular constructs featuring palladium coordination nodes. As this DS contribution arises from solvation, these ndings broadly reect the thermodynamic drive of self-assembly to form compact supramolecular structures. Furthermore, we demonstrate the utility of MD-based approaches to quantify the thermodynamics of large supramolecular systems, providing a methodology that enables in silico studies of self-assembly processes.

Synthesis and characterization of assemblies based on M and linL
Previously, we reported that the absence of trace halide impurities during the self-assembly of coordination cages resulted in slower formation kinetics, giving rise to the observation of intermediates. 49 Thus, we developed an alternative preparation for M, using a limiting quantity of palladium dichloride to minimize trace chloride (Scheme S1 †). With this prepared metal precursor, the self-assembly of stoichiometric quantities of  Fig. 1. Peaks corresponding to M 3 lin L 3 (d ¼ 9.24 ppm) and M 4 lin L 4 (d ¼ 9.40 ppm) were consistent with the reported values of these macrocyclic species. 3 Two additional peaks (d ¼ 9.32-9.37 ppm) present, with chemical shis consistent with reported oligomers, 1,2 and a single diffusion constant (D ¼ 1.56 Â 10 À10 m 2 s À1 , Fig. 1 inset). These features indicate the presence of a single coordination assembly with a size larger than free lin L (D ¼ 1.86 Â 10 À10 m 2 s À1 ) but smaller than M 3 lin L 3 (D ¼ 1.20 Â 10 À10 m 2 s À1 ). We also observed a near 2 : 1 ratio of a-pyridyl peak areas ( Table 1), assuming the overlap of a-pyridyl protons adjacent to coordination, we assigned these peaks to an oligomer species with the composition M 2 lin L 3 .
While previous studies rationalized that the self-assembly process is driven to maximize the number of coordination bonds formed, affording coordinatively saturated species that minimize the DH of the system. 1 However, analysis by Weilandt et al. on mononuclear Pd complexes demonstrated that the formation of successive coordination bonds result in diminishing DH contributions to the DG of complex formation, which was partially compensated by DS. 56 Our observation of a significant presence of oligomeric (coordinatively unsaturated) assemblies (14.9%, Table 1), we infer that DS may play a similarly signicant role in macrocycle assembly.

Thermochemical analysis of macrocycle-oligomer equilibria
Following the literature, we employed VT-NMR to quantify the relative abundance of assemblies (M 2 lin L 3 , M 3 lin L 3 , and M 4 lin L 4 ) by monitoring the intensity of their unique a-pyridyl peaks (Table 1) over a wide range of temperatures (297.5-350.0 K, see Fig. S12-S16 †). 5 To determine the relative DG of M 2 lin L 3 , M 3 lin L 3 , and M 4 lin L 4 we modeled the system as three orthogonal equilibria between each product and a common pool of reactants (Scheme S2 †). These relative DG values (Table S1 † exhibits an internal strain relative to M 2 lin L 3 manifesting as the DH-difference between the two complexes (DDH ¼ À2.27 kcal mol À1 ). As both assemblies are presumed to adopt a conformation where :N-Pd-N ¼ 90 , the apparent DDH is  Table 1.    (Table 1) with a simplified reaction model (Scheme S2 †).  (Fig. 2). 58 not accounted for when a simple geometric rationale is invoked. 1 Moreover, DS-differences between the 6-component M 3 lin L 3 and 5-component M 2L3 contrast the typical rationale, which correlates the integration of fewer components to a favorable DS. These ndings highlight how the current rationale for determining DS and DH contributions is insufficient to account for oligomeric assemblies, necessitating further computational investigation into the origins of internal strain found in M 4 lin L 4 and the unexpected DS penalties associated with M 2 lin L 3 formation.

MD analysis of experimental assemblies
Weilandt et al. suggested that two discrete DS factors that exert signicant inuence on metal-organic complex formation. The rst is DS struct , which decreases as more molecules (i.e., components) are required to form a complex. 5 The other is DS solv , which decreases as more solvent molecules are required to solvate a complex. 56 Using a previously described methodology, 52 we developed parameters to simulate assemblies comprised of M and lin L with accurate DH contributions (Fig. S4 †). Using these parameters, and GBSA model solvation, 54 50 ns trajectories ( Fig. S5-S7 †) were propagated by MD for M 2 lin L 3 , M 3 lin L 3 , and M 4 lin L 4 assemblies (Fig. 3). These trajectories were then used to compute the DS struct, 53 DS solv , 54 and DH contributions to DG ( Table 3).
The thermodynamic contributions to DH and DS computed from these simulations (Table 3) differ those obtained by VT-NMR (Table 2) in their absolute value. However, the differences (i.e., DDG, DDH, or DTDS) between assemblies measured by simulation and experiment are very similar (see below). The differences in absolute value reect the different reference states in experimental and computational measurements (Fig. S20 †). The reproduction of the relative differences in these physical quantities validates our in silico methodology for the thermodynamic values of these and similar assemblies.
The DH difference between M 2 lin L 3 and M 4 lin L 4 measured by VT-NMR (DDH exp ¼ +2.28 kcal mol À1 , Table 2), is similar to our MD-derived results (DDH MD ¼ +2.20 kcal mol À1 , Table 2). As DH generally originates from molecular geometry, we infer that the M 4 lin L 4 adopts a geometrically unfavorable (i.e., strained) conguration compared to M 2 lin L 3 . Visualization of MD trajectory data for M 2 lin L 3 assemblies reveals that this oligomer prefers a zig-zag conformation with a near-ideal square-planar coordination geometry at the Pd center (:N-Pd-N ¼ 89 , Fig. 3c). In contrast, visualization of M 4 lin L 4 reveals a foldedsquare structure that features a hyperbolic geometry (i.e., :N-Pd-N ¼ 86 , Fig. 3b) giving rise to an internal strain that is enthalpically unfavorable (i.e., elevates DH). Additional simulations of M 4 lin L 4 (performed in vacuo) reinforce that these distortions are a consequence of the solvation incurred to minimize the solvent-accessible surface area (Fig. S8 †) Table 3. Adjacent to each model is the average :N-Pd-N observed during MD.  (Table   3). This leads us to infer that experimental ÀTDS penalties associated with M 2 lin L 3 formation originate from these DS solv contributions, in agreement with thermodynamic studies of mono-nuclear Pd complexes. 56 These thermodynamic parameters demonstrate that DS-specically DS solv -drives the conversion of oligomeric intermediates (i.e., M 2 lin L 3 ) to their macrocyclic product assemblies (i.e., M 3 lin L 3 and M 4 lin L 4 ). Moreover, the effect of DS solv may overcome DH contributions, resulting in strained and distorted molecular geometries. While chemists have previously exploited DH to direct the formation of desired constructs, these ndings reveal that DS ultimately drives the synthetic process of multi-component self-assembly.

MD modeling of arbitrary M nLn and M nLn+1 assemblies
Oligomers similar to (and including) M 2 lin L 3 are theorized to form as intermediates in the self-assembly process of larger supramolecular structures (Scheme 1). Therefore, we utilized our MD-based approach to compare a range of oligomeric intermediates (M n lin L n+1 ; n ¼ 1-28) and the potential macrocyclic products (M n lin L n ; n ¼ 2-28) to elucidate the role of individual thermodynamic parameters (DS solv , DS struct , and DH) on DG for the self-assembly of macrocycles (Fig. 4).
The resulting simulations reveal that both macrocyclic (Fig. 4a) and oligomeric (Fig. 4b) assemblies exhibit increasingly unfavorable DG with increasing assembly size driven by DS struct contributions. The limited range of assemblies observed by NMR measurements (i.e., M 2 lin L 3 , M 3 lin L 3 , and M 4 lin L 4 ) is rationalized by the elevated DG experienced for other possible structures. This outcome is consistent with ESI-HRMS analysis (Fig. S9 †) that provides qualitative evidence for the existence of larger assemblies in low abundance (i.e., low signal-to-noise). The value of ÀTDS struct increases with size ( Fig. 4, blue trace) for the self-assembly of both oligomeric and macrocyclic products, consistent with the decreased degrees-of-freedom experienced upon aggregation. 57 Intriguingly, we nd a nonlinear correlation between the size and DH of oligomeric assemblies that is absent for macrocyclic congeners. Visualization of MD trajectory data reveals that larger oligomer assemblies adopt a compact conformation (Fig. S10 †), resulting in increased strain (i.e., DH penalty) on the palladium-pyridyl bonds compared to the zig-zag conformation found in smaller assemblies such as M 2 lin L 3 (Fig. 3c). We infer that these compact suprastructures are necessary to realize a more compact assembly, akin to the folded structure observed for M 4 lin L 4 macrocycles (Fig. 3b). This trade-off between ÀTDS solv and DH originates from solvation and distinguishes oligomeric assemblies from macrocyclic ones. As ÀTDS solv favors the formation of compact suprastructures, it is reasonable to deduce that the self-assembly of product macrocycles, in general, is driven by DS solv contributions.
Our simulations reveal the Pd 12 ben L 24 exhibits a lower DH (i.e., minimal geometric strain) compared to congeneric assemblies (Table 4). This observation is consistent with the literature and originates from the decreased metal-ligand bond strain experienced by this particular assembly-conguration. [8][9][10][11][12][13]49,52 Models of partially-formed assemblies (e.g., Pd 5 ben L 14 ) bear an elevated DH (Fig. 5a, blue trace) as a result of strain originating from deation or collapse during MD simulations (Fig. S11 †). Parallel observations have been made in M 4 lin L 4 macrocycles (Table 2), inferring increases in strain can act to offset penalties from solvation entropy (i.e., DS solv ), which leads to an overall elevation in DH for the system. The sum of entropic contributions (i.e., DS struct + DS solv ¼ DS, Fig. 5a, green trace) suggests that the formation of early intermediates (n <7) is hindered while the self-assembly of spherical cages (n ¼ 12) is encouraged. These results demonstrate that while DH directs the polyhedral geometry of the nal assembly, 52,55 DS drives the structure of self-assembly to be spherical. The comparison of the free energy (DG) pathways for the selfassembly of different topologies (Fig. 5b)

Conclusion
The thermochemical analysis of the self-assembly processes in palladium-based coordination macrocycles revealed unexpected DS-contributions that drive the formation of higher-order macrocycle assemblies (M 3 lin L 3 and M 4 lin L 4 ) from oligomer intermediates (M 2 lin L 3 ). Using an MD-based approach, we found that the driving force for self-assembly originates from the solvation entropy (i.e., DS solv ) of oligomeric intermediates that effects surface-area minimization of the construct. Thermodynamic trends were established by MD analysis of larger assemblies, revealing that both DS solv and DS struct direct the formation of assemblies that exhibit similar DH. Data from MD models of formation pathways for palladium-based coordination cages reveal that DS solv is responsible for driving the self-assembly process. Further application of our MD approach enables rationalization of the formation Pd 12 ben L 24 cage products over kinetically trapped congeners (i.e., Pd 8 ben L 16 ) directly from the computed thermodynamic quantities (DS solv , DS struct, and DH) of the intermediate assemblies. Overall, these complementary experimental and computational investigations expose DS as the driver for the formation of these desirable highly ordered structures that have broad applications across supramolecular chemistry.

Author contributions
DAP conceived, designed and performed the experiments. EOB contributed ligand materials, performed HR-MS analyses and provided experimental expertise for self-assembly experiments. JNHR and SM supervised the work. DAP, SM, and JNHR wrote  the manuscript with all authors providing signicant contributions to the analysis and interpretation of the work.

Conflicts of interest
There are no conicts to declare.