The building procedure for the Au18S14 gold nanoparticle, functionalized by the negatively charged 4-mercaptobenzoate ligand
The sulfurs were bound covalently to the gold at the surface. A copy of the ligand was made for each of the sulfurs and they were arranged around the core in the vacuum. Afterwards the energy of the system was minimized in the vacuum until all the bonds acquired their minimum-energy lengths and angles according to the CHARMM force field.
As the particles become smaller, we can expect the shapes to become less and less spherical, which decreases the surface curvatures on certain sides, thus allowing for the formation of more salt bridges in solution. On the other hand, since the binding can only occur on those sides with smaller curvatures, an entropic repulsion can be expected to appear due to the restricted number of possible orientations. Additionally, if we keep the shape constant, then by decreasing the size of the particle the curvature of the surface will increase. This in turn decreases the contact area between two particles, thus hindering the formation of salt bridges between them. For our full explanation on the effects of size and shape please see this reference.
The Au144S60 gold nanoparticle functionalized by the negatively charged 11-mercapto-1-undecanesulfonate ligand, placed in near-physiological conditions
The nanoparticle was solvated in a 10x10x10 nm water box and ionized by NaCl with a concentration of 150 mM. The temperature was kept at 298 K and the pressure at 1 bar in order to achieve near-physiological conditions. Periodic boundary conditions were introduced in every direction, i.e. the system is replicated infinitely so that whenever a particle crosses the boundary on one side of the unit cell, it will return from the opposite side. The system contained around a hundred thousand atoms.
The bound state of a pair of these particles corresponds to the formation of a monolayer of counterions located in-between them, through which salt bridges are formed. In this reference we compare the effects of choosing 5 different types of ligands, with different properties of length, sing of the charge and flexibility. The binding is stronger for shorter ligands since they fluctuate less and thus the counterions are able to remain bound to them more easily as the particles approach. Under physiological conditions, negative particles present stronger attraction due to sodium binding more strongly to the terminal groups and having more room in-between the particles. Flexible ligands fluctuate more strongly and thus weaken the bonds with counterions; however, when the particle is very small (and thus the amount of ligands is scarce), this flexibility also facilitates the formation of salt bridges.
The Au144S60 gold nanoparticle functionalized by the negatively charged 4-mercaptobenzoate ligand, placed in 5 different aqueous environments
In this reference we compare the effect that various shapes, sizes and concentrations of hydroxide ions have on the interactions of gold nanoparticles. The higher concentrations of electrolyte increase the aggregation tendencies, since they increase the density of the counterion cloud at the middle. As the particles approach, various interactions emerge which attempt to extract the counterions from the region in-between them: an entropic effect originating form the counterion confinement in a smaller region, an electrostatic repulsion between the counterions and the partial dehydration and steric collisions of the counterions as the two counterion shells overlap with each other. The steric collisions of big counterions are significant enough to cause repulsion between the particles. Smaller counterions facilitate the aggregation of particles the most because of their higher affinity to the surface and because they displace the least amount of water from the surface.
Polarized counterions such as TRIS cause the transition from the near-zero net interaction at the long range to the strong repulsion at the short-range to occur very suddenly. Even though this ion was the biggest under study (which should thus intensify the effect of steric collisions), the charge polarity increased its binding strength to the surface, thus allowing it to remain bound despite the collisions. We also studied the effect of pH, and observed that a lower net charge of the particles caused the net interparticle interaction to become repulsive (the aggregation state becoming meta-stable) due to the smaller amount of salt bridges binding the particles together. In the pure-water case the repulsion is about five times stronger than in the other cases.
The dissociation energy of a pair of Au102S44 nanoparticles functionalized by the negatively charged 4-mercaptobenzoate ligand, obtained through constant-velocity, straight-line external forces
The Potential of Mean Force (PMF) is computed through a multi-sectional scheme by steering the system in small steps from state A to B and then back to A four times, allowing it to equilibrate after every step for 1 ns. Since these particles are near-spherical, the Euler angles defining their orientation and the spherical angles defining their angular position can all be integrated out and the PMF becomes one-dimensional as a function of center-to-center separation.
When the particles are non-spherical (as is the case of Au18S14), this one-dimensional PMF approximation cannot be used to obtain the dissociation constant. It becomes necessary to sample over the orientational and positional degrees of freedom (the conformational degree of freedom is not relevant since the particle is rigid). For this purpose, the fluctuations of three atomic groups per particle are sampled at both the bound and unbound states. The entropic contribution of this bound/unbound fluctuation difference to the dissociation constant is obtained by considering the fluctuations to have a Gaussian distribution.
The dissociation energy of a pair of Au18S14 nanoparticles functionalized by the negatively charged 4-mercaptobenzoate ligand, obtained by restraining the geometrical degrees of freedom through external biasing potentials
The PMF is computed through a procedure known as Adaptive Biasing Force, whish measures the mean force acting on the particle and cancels it out, so that all the energy barriers disappear and the particles are free to undergo Brownian motion, in this way achieving a uniform sampling over the phase space. The dissociation is separated into multiple intermediate stages, each of which consists of applying geometrical restraints to each of the degrees of freedom of the ligand through a biasing potential. For this purpose, the ligand is assigned seven degrees of freedom: the root mean squared displacement which defines its conformation, the three Euler angles which define its orientation, and the three spherical coordinates which define its position. These values are defined in terms of six atomic groups: three from one particle and three from the other. A PMF is computed for each of the degrees of freedom.
The restraining potentials are summed to the internal energy of the system, and the energetic cost of applying the restraint is obtained by measuring the fluctuations of the degrees of freedom from the lowest-energy point of the restraint. Both the conformational and rotational components of the dissociation energy favour the unbound state, where the entropic freedom is larger. However, the translational component favours the bound state, and it is larger than the other two combined.