of low-valence DNA nanostars : a colloidal model for strong glass formers †

Kinetic arrest in colloidal dispersions with isotropic attractive interactions usually occurs through the destabilization of the homogeneous phase and the formation of a non-equilibrium network of jammed particles. Theory and simulations predict that a different route to gelation should become available when the valence of each colloidal particle is suitably reduced. Under these conditions, gelation should be achievable through a reversible sequence of equilibrium states. Here we report the reversible dynamic arrest of a dispersion of DNA-based nanoparticles with anisotropic interactions and a coordination number equal to four. As the temperature is decreased, the relaxation time for density fluctuations slows down by about five orders of magnitude, following an Arrhenius scaling in the entire experimentally accessible temperature window. The system is in thermodynamic equilibrium at all temperatures. Gelation in our system mimics the dynamic arrest of networking atomic strong glass formers such as silica, for which it could thus provide a suitable colloidal model.

Colloidal dispersions have played a crucial role in understanding fundamental problems in condensed matter physics.In particular, extended and continuous efforts have been devoted to the study of the slowing down of the dynamics on approaching the glass and the gel transition.Colloids dominated by repulsive, excluded volume interactions freeze into non-ergodic states at large concentrations, providing a colloidal counterpart of fragile glass-forming atomic and molecular liquids. 1 It is much harder to devise colloidal models of liquids forming the so-called "strong glass" state. 2 Materials belonging to this class, whose archetype is silica, are characterized by atoms or molecules forming networks with a low coordination number and by an Arrhenius slowing down of internal modes with temperature.Deformable repulsive spherical colloids, found to display an Arrhenius dynamics in concentration, 3 miss most of the relevant features of these molecular systems: network, attractive interactions, and Arrhenius temperature dependence.
Colloids interacting via short-range attractive potential (e.g.depletion) can undergo kinetic arrest entering a state in which the particles form an extended network, 4,5 generally referred to as a gel state.6][7][8][9] During the phase separation following the quench, colloids condense into an interconnected network of locally dense glassy regions.The system eventually freezes into a disordered heterogeneous gel which retains the structural pattern imposed by the interrupted spinodal decomposition. 8espite their network structure, this route to gelation has little in common with network-forming atomic glasses since the dynamic arrest is here obtained through a discontinuous transition to a non-equilibrium state and since the jamming is provided by the tight packing and large coordination number of the interacting particles.
Phase-separation mediated gelation is however only one of the mechanisms expected to generate a colloidal gel.Recent numerical and theoretical studies 10,11 have suggested the possibility of a different route available when colloidal particles can interact with a limited number of neighbours only.1][12] Cooling in this range of c, where the only arresting mechanism is the increasingly large lifetime of inter-particle bonds, is expected to lead to a kinetically arrested homogeneous gel through a continuum of equilibrium states with intermediate dynamical response. 12his path toward gelation, controlled by the bond lifetime rather than by excluded volume interactions, is analogous to the dynamic arrest that characterizes "strong" network glass This journal is © The Royal Society of Chemistry 2015 Soft Matter PAPER formers, 13 molecular systems for which they could provide the colloidal counterpart.
Although various strategies for synthesizing colloidal particles with specic valence have been developed in the last few years, [14][15][16] the production protocol of these patchy particles is far from yielding the bulk quantities that would be necessary to explore their collective thermodynamic and dynamic behavior.8][19] This strategy enables the production of large amounts of identical nanoparticles with controlled valence and interaction strength, [20][21][22][23] suitable for the investigation of the phase behavior. 23n this article we report the rst experimental study of the dynamical changes accompanying equilibrium gelation in a system of well-dened valence DNA-nanostars (NS), whose phase-diagram was previously determined in ref. 23.To this aim we selected concentrations large enough not to incur in liquidgas phase separation.At such concentrations, if no other phase transition occurs, the system is expected to be homogeneous and stable at all temperatures (T) down to water freezing.Our data conrm that these systems evolve from a homogeneous "uid" state of free NS in solution at high T to a homogeneous dynamically arrested, highly viscous state at low T, in which a macroscopically extended network of strongly bound NS is formed.Such transition is found to be continuous, with the "uid" of DNA-NS progressively slowing down toward a dynamically arrested gel through a reversible sequence of equilibrium states.

DNA nanostars
Low valence colloids are obtained via self-assembly of DNA single strands of suitably chosen sequences (see Methods) at temperatures lower than z65 C. Each particle results from the aggregation of four distinct strands and takes the shape of a NS with four arms (of length ' z 8 nm) terminating with identical overhanging unpaired sequences that provide the NS-NS interaction (Fig. 1a).The overhangs, which become sticky at temperatures lower than z40 C, are made of 6-base long self-complementary single strands (CGATCG).Unpaired bases are inserted by sequence design just before the overhang sticky sequence as well as in the particle core, so that the angle between the arms of a NS can uctuate, their relative distance being controlled mainly by steric and electrostatic repulsion.Previous combined experiments of optical microscopy, UV absorbance and light scattering, devoted to determine the phase behavior of this system, 23 demonstrated the presence of an unstable region in the c-T space where the aqueous dispersion separates into a "gas" and a "liquid" of NS.Fig. 1b shows the binodal line for such a transition, obtained from concentration measurements of coexisting gas and liquid phases.At low T, the density of the coexisting liquid was found to be c l z 17 AE 1 mg ml À1 .In line with theoretical predictions, 11 c l is comparable to the minimum density of fully bonded networks, in which each particle is surrounded by four neighbours.Such a density is much smaller than that of a "liquid" of isotropically interacting colloids, whose number of neighbours is typically much larger (z12).Concentrations c T c l , at which no phase separation takes place on cooling, can be accessed experimentally without incurring into experimental issues such as large viscosity or mis-shaped structures, or packing limitations, making DNA NS an ideal system to explore the features of reversible equilibrium gelation.To investigate equilibrium gelation we thus prepared samples at two DNA NS concentrations, c 1 z 17.5 mg ml À1 ($c l ) and c 2 z 21.3 mg ml À1 (>c l ) and studied their behavior as a function of temperature.The sample preparation procedure (see Methods) ensures that the c 1 sample does not cross the phase boundary in the explored range.Inspection of samples by optical microscopy conrms that they remain homogeneous and clear, with no sign of phase separation, in the whole T range.
As observed from investigation of colloids with DNA-mediated attractive interactions, 24 inter-particle interactions provided by DNA pairing are particularly convenient to investigate gelation because of the large entropic component of the hybridization free energy, which makes the inter-particle interactions strongly T dependent.This can be appreciated by inspecting the right axis of Fig. 1b, reporting the ratio between the bonding free energy DG nn ¼ À44.6 + 0.13T kcal mol À1 involved in each overhang interaction (as calculated according to the standard nearest-neighbor model 25 ) and the thermal energy RT (where R is the gas constant).In the temperature range examined in this work, which spans 35 C, the ratio increases by over a factor of two.Such a large span of DG nn /RT cannot be obtained with more conventional particles where the attractive interaction has a weaker dependence on T.

Light scattering
To investigate the homogeneity of the samples on mesoscopic length-scales, we measured the dependence of the scattered intensity I(q,T) on wave-vector q and T. The small sample volumes employed in this study (z50 ml) make it hard to precisely evaluate the effective scattering volume, given by the overlap among the illuminating beam, the collected modes and the actual cell.Therefore, while the determination of q itself and the dynamics are accurate, we have less precision in the measurement of the q dependence of the scattered intensity.Within our experimental accuracy, no signicant q dependence was observed (see Fig. S2 of the ESI †), indicating that at the q values accessible by light scattering (q ( 2p/') the system remains nearly homogeneous at all the explored T. Such behavior marks a substantial difference from the one observed in the critical system, where concentration uctuations develop on the mesoscale (see ref. 23 and Fig. S2 of the ESI †).Fig. 2a shows I(T) measured along the thermodynamic paths indicated by the two vertical lines in Fig. 1b (continuous at c 1 and dashed at c 2 ).In both systems, I(T) exhibits a mild increase on cooling, but no sign of incurring spinodal instability or a critical point which would lead to a singular behavior such as the one found along the critical isochore (dotted line in Fig. 1b) in ref. 23, here reported for comparison (diamonds in Fig. 2).The fact that the growth of I(T) observed upon lowering T in the c 1 sample is larger than that of the c 2 sample may reect its closer proximity to the spinodal line.
Dynamic light scattering measurements performed along the same paths reveal instead a strong change in the kinetic behavior.Fig. 1c shows a selection of the collective intermediate scattering functions g 1 (s) measured at various T for c 1 (analogous results can be found for c 2 in the ESI †).While at high T a single process dominates the dynamics, upon cooling a twostep relaxation develops, the slower process undergoing a very marked and progressive slowing down.
As previously proposed for gels, 26,27 we quantify the collective dynamics of the system by representing g 1 (s) as a sum of an exponential and of a stretched exponential decay, modeling the fast (f) approach to the plateau A s and the slow (s) relaxation to zero, respectively.This functional form is also commonly used to interpret the decay of correlations in systems approaching dynamic arrest, 13 including transient networks. 28or all T and all q the functional form in eqn (1) provides a quite accurate description of the decay of the density uctuations, as can be seen in Fig. 1c.Since the stretching exponent b s was found to consistently assume values between 0.6 and 0.7, we xed it to b s ¼ 0.7 to provide a uniform analysis of all the relaxation curves.

Testing thermodynamic equilibrium
Given the relevance of determining whether the DNA-NS system is in thermodynamic equilibrium in all the explored T range, we carefully examined this aspect.An important indicator is offered by the relative mean squared intensity uctuation r ¼ hI 2 i/hIi 2 , obtained from the ratio of the s ¼ 0 and s ¼ N values of the correlation function of the scattered intensity.A typical effect of the loss of ergodicity is the decrease of such a ratio and its dependence on the specic position where the scattered intensity is collected, which indicates a growing static component in the scattered intensity.Here instead, r does not decrease with decreasing T and it remains constant upon rotating the sample in the light scattering apparatus.This evidence supports the notion that for t > s s all uctuations in c are fully relaxed, which is by itself a clear indication of ergodicity in the length scales probed by light scattering.
To further strengthen this conclusion, and rule out the presence of possible metastable states, we thermally cycled between 10 and 45 C one of the samples at c 1 along various T paths.Measurements at two target T (15 and 20 C) were taken both by slowly ramping T (by about 1 C h À1 ) and aer fast quenches (of about 1 C min À1 ).The results are shown in Fig. 3, where we represent the g 1 (s) measured aer various thermal paths (panel a) together with the corresponding tting parameters (panel b).Within our experimental accuracy we could not detect any dependence of the thermalized states on their previous conditions.We thus conclude that along the explored path our data always refer to equilibrium uid states.

Approach to gelation
The T and q dependence of the t parameters is shown in Fig. 4a and in Fig. 5 for both c 1 and c 2 .The characteristic time of the fast decay, s f (Fig. 4a, full dots), is very weakly dependent on T while it shows an appreciable dependence on q.Fig. 5a shows that s f data are compatible with a q À2 dependence.In contrast, the characteristic time of the slow relaxation, s s (empty dots), is found to be q independent, as shown in Fig. 5b.Both s f and s s are nearly identical for c 1 and c 2 , indicating that the dynamic behaviors of these gelling systems do not depend signicantly on c.
The T dependence of s s , shown in Fig. 4a, is characterized by a neat activated-type behavior, as evident by the agreement between data and the dashed line, representing the best tting Arrhenius equation s s ¼ s 0 exp(DG(T)/RT), where DG(T) is the variation of Gibbs free energy involved in the activation process and s 0 is the high T limit of s s .We previously found an analogous Arrhenius slowing down in a more restricted T range studying samples prepared at the critical concentration. 23In that case, only T > 25 C could be investigated before entering the coexistence region, a constraint that forbade exploring gelation in the absence of phase separations.Here instead, samples can be cooled down to freezing without phase-separation.At both concentrations c 1 and c 2 , the system is characterized by a growth of s s of more than 5 orders of magnitude, from 100 ms to over 10 s, marking an impressive dynamic arrest under equilibrium conditions.Such kinetic behavior coherently combines with the data in Fig. 2 to indicate that in the whole range up to gelation the system smoothly evolves with no structural or dynamical singularity.The independence of the relaxation time on c strongly suggests that temperature, as opposed to concentration, controls the relaxation time, generating in the equilibrium T-c gel phase isochrones parallel to the concentration axis. 12Fig.4a displays ln(s s ) vs. (RT) À1 to enable extracting DH from the slope, while the intercept contains combined information on s 0 and DS.From the simultaneous t of data at different q we nd DH z 58 AE 5 kcal mol À1 .Such a value indicates that the free energy barrier of the activated relaxation process is comparable with the enthalpy of 1.3 overhang bonds as estimated by the nearest-neighbour model. 25Assuming DS ¼ 170 cal mol À1 K, the value from ref. 25  for 1.3 bonds, it is possible to estimate the attempt time s 0 from the intercept of the line ln(s s ) vs. (RT) À1 , obtaining s 0 z 1 ms.The estimate of 1.3 bonds has some factors of uncertainty, including the differences in hybridization enthalpy when computed according to the various values of the thermodynamic parameters found in the literature, 19 the stabilizing effect of the presence of bases adjacent to the paired 6-mer, 25 and the destabilizing effects of long DNA sequences connected to paired ones. 29While all these factors make it impossible to obtain a secure determination of the activation energy as a multiple of the single inter-NS bond enthalpy, the data unambiguously indicate that the activation energy is of the same order of such DNA-DNA bonds.
These results suggest that once the network is formed, the system dynamics is enslaved to the activated elementary step of bond-breaking.Why the energy involved in the network rearrangement is approximately 1.3 bonds and not a larger value  The notion of kinetic arrest through the progressive formation of an extended network of bound DNA-NS is supported by the behavior of the viscosity of the system, the other typical quantier of the dynamical arrest, which we could measure in the case of the c 1 system (see the ESI †).Indeed, we nd that the viscosity depends on T through an Arrhenius behavior with equal DH.The approach to kinetic arrest through an equal Arrhenius law in both relaxation rate of spontaneous uctuations and in viscosity is indeed the dening property of strong glass formers, 13 compounds typically having a limited coordination number and forming arrested phases at reduced densities lower than those of the fragile glass formers.All these features are captured by the DNA-NS system investigated here.
The observed q-independence of s s and q À2 dependence of s f are features that are remarkably shared by other physical gels. 28he q-independence of s s suggests that the network rearrangements that follow bond breaking events equally contributes to all length scales probed in our experiments.This feature could be produced by relaxations that act over distances much larger than the nearest neighbour distances.The q À2 dependence of s f indicates a diffusive process that maintains the same time scale across the explored temperature window, over which the system moves from a collection of freely diffusing nanostars (or small aggregates) to a fully bonded network where all particles are connected in the same cluster.Consistently, s f quanties how the collective density uctuations decay over length scales signicantly larger than the nearest neighbour distance due to the free diffusion of small aggregates at high T and the diffusive density uctuations of the constrained network structure at low T.This latter behavior could be analogous to the concentration uctuations in polymer gels due to mesh uctuations.

Non-ergodicity factor
The activated slowing down observed here is the consequence of the remarkably long lifetime of oligomer DNA duplexes and of its dramatic T dependence, 30 features that already enabled unveiling of unexpected self-assembly processes otherwise hard to detect. 31f the bond lifetime were innite as in chemical gels, the network would not restructure and the long time limit of g 1 (s) would coincide with A s . 32,33Here instead the non-ergodic behavior is resolved by the activated unbinding of the NS overhangs, which reestablishes the access to the whole conformational space for s > s s .The two-step behavior of the correlation function is characteristic of systems continuously approaching a kinetic arrested state and it is also commonly observed in supercooled glassforming liquids.However, different from the glass case, in which the plateau height remains constant-suggestive of a localization length that does not change on supercooling until the (ideal) glass transition is reached 34 -here we observe a clear growth of the plateau height (commonly indicated in the glass community as the non-ergodicity factor 13 ), correlated with the slowing down of the dynamics.
As can be appreciated from Fig. 4b, the non-ergodicity factor A s signicantly grows with cooling, progressively increasing from zero at high T, where NS are not interacting, to z0.9 (for c 1 ) and z0.8 (for c 2 ) at the lowest investigated T. The amplitude of A s does not depend on q in the explored hydrodynamic region.Previous theoretical 35 and numerical 32,36,37 studies of the selfdynamics in chemical gels (innite bond lifetime) have associated the emergence of non-ergodicity to the percolation transition with the non-ergodic component of g 1 (s) continuously growing with the fraction of particles frozen in the innite spanning clusters.In analogy, we interpret the growth of A s in our collective intermediate scattering functions as a consequence of the network consolidation caused by the progressive attachment of the NS to the percolating aggregate.Fig. 4b displays the fraction f of paired overhangs as a function of T, as calculated by following ref.38, which indeed traces the growth of A s in the entire T window.The smaller value of A s observed at c 2 could perhaps indicate that at concentrations above that of an ideal crystal, at which f ¼ 1, the fraction of bound overhangs decreases with c.
The fortunate combination of bond strength and lifetime enables us to observe and characterize the two step relaxation already starting from low values of A s , approximately corresponding to f z 1/3, the mean-eld estimate for the fraction of formed bonds yielding percolation in a system of particles with valence four. 39

Conclusions
By exploiting the programmability of the interactions among DNA nanostars, we have provided the rst experimental dependence, for comparison.s f is found to be T independent and q dependent, while s s is T dependent and q independent.demonstration of reversible equilibrium gelation in limited valence particles.On lowering T, the system progressively evolves into a kinetically arrested state through a sequence of equilibrium states characterized by an increasingly slow dynamics without incurring into phase transitions.Both the density relaxation time and the viscosity display an Arrhenius dependence on T, a behaviour corresponding to the onset of the glass state in network glass-forming liquids, 13 providing the rst colloidal model for strong glasses.However, different from this class of atomic and molecular network-former systems, here the non-ergodicity factor crucially depends on T. We attribute this to the fact that in atomic systems the bond lifetime is so short that the two-step relaxation becomes detectable only when the network is fully formed.Here instead the inter-NS bonds are already long-lived close to percolation, a condition enabling us to detect the equilibrium pathway toward gelation, inaccessible in ordinary glass-forming liquids.
Our observation of a continuum of transition into the gel state by cooling samples having concentrations larger than the instability region agrees with recent studies on the thermodynamic stability of tetrahedral coordinated particles showing that bond exibility, certainly present in the DNA-NS system, entropically stabilizes the disordered, fully bonded structure, rendering the gel more stable than any crystal phase. 40,41The experimental evidence of the absence of crystallization-despite the investigated sample concentration being the same as that of a tetrahedral open crystal (diamond) and despite the several months' shelf life of our samples-points in this direction.
Overall, our results conrm and extend the relevance of colloids as models for the investigation of the glass state.As hard-sphere colloids were demonstrated to be a paradigmatic system to explore fragile glass formers, [42][43][44] DNA-based directional colloids will possibly contribute to the understanding of the glass transition in strong glasses.

Sample preparation
DNA NS results from the self-assembly of four 49-base long DNA oligomers.Sequences are as follows: Sequence 1. 5 0 -CTACTATGGCGGGTGATAAAAACGGGAAGAG CATGCCCATCCACGATCG-3 0 Sequence 2. 5 0 -GGATGGGCATGCTCTTCCCGAACTCAACTGC CTGGTGATACGACGATCG-3 0 Sequence 3. 5 0 -CGTATCACCAGGCAGTTGAGAACATGCGAGG GTCCAATACCGACGATCG-3 0 Sequence 4. 5 0 -CGGTATTGGACCCTCGCATGAATTTATCACC CGCCATAGTAGACGATCG-3 0 Each strand is designed to bind to two other strands in two 20-nucleotide-long regions (underlined letters), which form the arms of the structure.Each arm ends in a single stranded 6-base long overhang (bold letters) of sequence CGATCG.Since the sequence of the overhangs is self-complementary, and being identical in all arms, it provides a sticky-type interaction between the nanostar tips.In-between the arm forming sequences, as well as before the overhang sequence, unpaired A bases were inserted in order to release angular constraints between the arms.Page puried DNA was purchased from PRIMM srl.Desalting was performed using Illustra NAP-10 columns (GE Healthcare).Samples of valence-4 DNA-NS were prepared at c 1 z 17.5 mg ml À1 and at c 2 z 21.3 mg ml À1 .DNA was hydrated using NaCl electrolyte solutions to yield a total estimated ionic strength, which also includes the dissociated ions, of z50 mM.The pH of the resulting solution was measured and found to be in the range 7-8.All samples were heated to 90 C and slowly cooled to 30 C in approximatively four hours to allow the formation of the NS.The sample at c 1 was prepared by exploiting phase separation: we initially prepared a dispersion at the critical concentration c* ¼ 9 mg ml À1 , cooled it at T ¼ 4 C and centrifuged in order to speed up the phase separation process.The vapor phase was then carefully removed and the procedure was iterated until no further phase separation was observed.

Light scattering
Static and dynamic light scattering measurements were performed with a customized light scattering setup (Scitech Instruments) equipped with a 532 nm solid-state laser source.Scattered light is collected by an optical ber carrying few propagation modes, yielding values of hI 2 i/hIi 2 z 0.5, for ergodic samples.Most of the data were taken at the scattering wave-vector q ¼ 22.3 mm À1 (q ¼ 90 ) as a function of T, in the range 10 C # T # 45 C. For each T three sets of acquisition spaced by two hour intervals were run.Additional data, with less statistics, were taken at q ¼ 12.1 mm À1 , q ¼ 17.6 mm À1 , and q ¼ 30.6 mm À1 .All explored q are well into the hydrodynamic regime.

Fig. 1
Fig. 1 (a) Pictorial view of a valence-4 DNA NS resulting from the selfassembly of four DNA oligomers.Inter-NS interactions are provided by the hybridization of the 6-base-long overhangs on the arm tips.(b) Phase diagram of DNA nanostars with valence-4 from ref. 23.Red symbols and orange dashed line mark the gas-liquid binodal line.The vertical lines indicate the valence-4 NS concentrations explored in this work: c 1 (continuous line) and c 2 (dashed line).The dotted line indicates the critical isochore.The colored dots at c 1 mark temperatures at which the correlation functions in panel C were measured.(c) Intermediate scattering functions g 1 (s) vs. time s measured at selected temperatures, as indicated by color matching with panel B, at q ¼ 22.3 mm À1 .Lines represent the best fit according to eqn (1).

Fig. 2
Fig. 2 Total scattered intensity as a function of T for DNA NS samples at three distinct concentrations: c 1 ¼ 17.5 mg ml À1 (red dots), c 2 ¼ 21.3 mg ml À1 (green squares), and c* ¼ 9.0 mg ml À1 (black diamonds).Data at the critical concentration c* are from ref. 23.

Fig. 3
Fig. 3 (a) Intermediate scattering functions g 1 (s) of the c 1 sample measured at T ¼ 15 C and at T ¼ 20 C after various thermal histories: slowly cooling from 45 C (black lines), slowly cooling from 45 C after holding the sample for 6 months at 4 C (blue lines), slowly heating from 10 C (orange lines), and fast cooling from 45 C (pink lines).(b)Fast and slow characteristic times and non-ergodicity factor determined from g 1 (s) at the T sequentially adopted to thermalize the sample.

Fig. 4
Fig.4(a) Dependence on 1/T (bottom axis) and T (top axis) of the characteristic times s f (full dots) and s s (empty dots) measured at three scattering angles in the c 1 sample (red symbols), and at 90 in the c 2 sample (green symbols).s f is constant with T, while s s slows down by more than five orders of magnitude.The dashed line shows the best fit to all the data of the Arrhenius law ln(s s ) vs. (RT) À1 .(b) Amplitude A s of the slow relaxation process.A s increases on cooling, growing upon lowering T to z0.9 (c 1 ) and z0.8 (c 2 ).The full black line indicates the fraction f of paired bases (thus ranging from 0 to 1) as calculated by NUPACK38 for the sequence CGATCG, at the same concentration and ionic strength as in the experimental sample.

Fig. 5
Fig. 5 Dependence on q of the characteristic times for s f (pane a) and s s (pane b) DNA-NS in the c 1 sample (red symbols) and in the c 2 sample (green symbols).Data are shown for T ¼ 45 C (dots), 35 C (squares), 26 C (diamonds), and 10 C (triangles).The dashed line shows q À2dependence, for comparison.s f is found to be T independent and q dependent, while s s is T dependent and q independent.