Connecting Irreversible to Reversible Aggregation: Time and Temperature

Abstract

We report molecular dynamics simulations of a gel-forming mixture of ellipsoidal patchy particles with different functionality. We show that in this model, which disfavors the formation of bondloops, elapsed time during irreversible aggregation -leading to the formation of an extended network -can be formally correlated with equilibrium temperature in reversible aggregation. We also show that it is possible to develop a parameter-free description of the self-assembly kinetics, bringing reversible and irreversible aggregation of loopless branched systems to the same level of understanding as equilibrium polymerization.Several natural and synthetic materials, as well as biological structures, result from the self-assembly of elementary units into branched aggregates and networks [1,2,3,4,5]. This self-assembly process is receiving considerable attention in two fast-growing fields: supramolecular chemistry [1, 2, 3] and collective behavior of patchy and functionalized particles [6,7,8], among the most promising building blocks of new materials [9,10]. The process of formation of an extended three-dimensional network of bonds connecting independent molecules, proteins or colloidal particles, is named gelation and the resulting material a gel [11,12,13]. The ratio between the bond energy u 0 and the thermal energy k B T (where k B is the Boltzmann constant and T is the temperature) can be used to classify aggregation into two broad categories: for strong attraction strength (chemical case [11,14,15,16]), bond formation is irreversible and the number of bonds continuously grows with time. In the case of weak attraction strength (physical case [17]), bonds break and reform while the number of bonds progressively reaches its equilibrium value. In the latter case, the final structure of the system can in principle be predicted with equilibrium statistical mechanics methods.The value of the ratio u 0 /k B T separates the two classes. Conceptually, any model of physical aggregation may be turned into a chemical model by studying its properties following a quench to k B T ≪ u 0 . Similarly, applying temperatures comparable to the bond energy turns an irreversible aggregation model into a physical one. The idea of a close connection between irreversible and reversible aggregation is already contained in the early mean-field theoretical work of Stockmayer [14], considering the Smoluchowski's kinetic equations solved in the limit of absence of closed bonding loops. In Stockmayer's calculations, at any time t during chemical aggregation, the distribution of clusters of finite size k (N k (t)) is identical to that found following equilibrium statistical mechanics prescriptions, i.e., by maximizing the entropy with the constraint of a fixed number of bonds. The N k are commonly referred to as Flory-Stockmayer (FS) distributions. Later on, Van Dongen and Ernst [18] confirmed that the FS distributions are also solutions of the Smoluchowski's equations when bond-breaking processes are accounted for. According to these theoretical wo...

Equilibrium gels of limited valence colloids

“…In this respect, it could appear odd that Laponite gels have been considered as an example of equilibrium gels. This association has been supported by theoretical work [8,60,61] demonstrating that systems of limited valence particles -when rapidly brought to very small temperatures to simulate an irreversible aggregation process -evolve in time by progressively decreasing their effective temperature. Such a mapping between aging time and temperature (in equilibrium) has been invoked to interpret Laponite phase behavior in the context of limited-valence equilibrium phase diagram [58].…”

Electrostatics and aggregation: How charge can turn a crystal into a gel

“…6 We set λ = 1.2 in our calculations, motivated by the recognition that proteins typically possess interactions of short range. 6,[16][17][18] To these expressions we add the electrostatic free energies f es d of each d-dimensional assembly. We calculated these free energies using the Poisson-Boltzmann (PB) equation,…”

Generic model for tunable colloidal aggregation in multidirectional fields

“…12,[17][18][19][20][21] Such network-like aggregates are considered to be the underlying micro-structures of gels and have been intensively investigated in experiment and theory under equilibrium as well as non-equilibrium conditions. [22][23][24] In the latter, qualitatively different aggregation mechanisms can be identified, namely diffusion limited cluster aggregation (DLCA) 25,26 and reaction limited cluster aggregation (RLCA). 27 In the DLCA regime each particle collision leads to the formation of a rigid and essentially (on the timescale of the experiment) unbreakable bond with fixed spatial orientation.…”