# Monte Carlo Procedure

Monte Carlo simulations are performed according to the Metropolis algorithm in the canonical ensemble. In this method, successive "trial" chain configurations are generated to obtain a reasonable sampling of low-energy conformations [36]. After applying elementary movements that are randomly selected, the Metropolis selection criterion is employed to either select or reject the move. If the change in energy AE resulting from the move is negative, the move is selected. If AE is positive, the Boltzmann factor p, p = exp

is computed and a random number z with 0 < z < 1 is generated. If z < p, the move is selected. When z > p, the trial configuration is rejected and the previous configuration is retained and considered as a "new" state in calculating ensemble averages. That conformation is the one that is perturbed in the next step. The perturbation process is continued a specified number of times (a typical run requires several million perturbations) until the conformation is energy minimized and equilibrated.

To generate new conformations, the monomer positions are randomly modified by specific movements. These movements include three "internal" or ele mentary movements (end-bond, kink-jump, and crankshaft, respectively), the pivot, and the reptation, respectively (Fig. 1). The use of all these movements is very important in ensuring the ergodicity of the system as well as the convergence toward minimized conformations. One important challenge and problem consist of allowing the energy of the complex structure to be minimized gradually without trapping the structure in a local energy minimum. This problem is of particular importance when compact conformations have to be achieved or when large polyelectrolyte chains are considered owing to the fact that a few monomer-monomer contacts can lead to the formation of "irreversible" bonds that freeze the complex structure. Some MC refinements are thus necessary to overcome the formation of structures in local minima and increase the chances of success when sampling new conformations. Anneal MC can be used to gradually minimize the complex structures by altering the temperature from an initial temperature to a final one and vice versa, so as to increase the chances of success when sampling conformations. However, as this method requires a large amount of CPU time, it should be used for the formation of dense structures only. Another approach that we are using consists, at each step of the MC procedure, of using the most effective movements (internal, pivot, or reptation) to achieve a more rapid and total exploration of the configurational space of the chains [37]. To check the efficiency and validity of our model and algorithms, the formation of dense globules was first experienced by considering that the computational efficiency of MC simulations is known to decrease dramatically reptation algorithm pivot algorithm reptation algorithm pivot algorithm end-bond motion kink-jump motion crankshaft motion end-bond motion kink-jump motion crankshaft motion

FIG. 1 To generate new conformations, the monomer positions of the polyelectrolyte chains are randomly modified by specific movements. These movements include three "internal" or elementary movements (end-bond, kink-jump, and crankshaft, respectively), the pivot, and the reptation.

FIG. 1 To generate new conformations, the monomer positions of the polyelectrolyte chains are randomly modified by specific movements. These movements include three "internal" or elementary movements (end-bond, kink-jump, and crankshaft, respectively), the pivot, and the reptation.

with system compactness. To generate a collapsed chain reptation is the most efficient algorithm to rapidly increase, in a first step, the degree of chain compactness. Then to achieve dense and collapsed spherical conformations, as reptation acceptance rate is poor, the combination of reptation and internal movements gives good results. On the other hand, to obtain extended structures, pivot movements are the more efficient and can be used alone. To achieve a necklace structure, i.e., a global stretched conformation composed of compact beads, the chain must first pass through a stretched structure. This is rapidly achieved, for example, with the pivot algorithm. A combination of reptation and internal movements is then necessary to form beads along the chain backbone. It should be noted that to sample the self-avoiding walk chain's conformational space, the three algorithms have a good acceptance rate. However, the pivot appears to be the most efficient procedure to sample new conformations, whereas if internal movements only are employed, a large number of conformations are required to eventually achieve a significant root mean square difference with the starting reference conformation.

To investigate the formation of polyelectrolyte-particle complexes, the central monomer of the chain is initially placed at the center of a large three-dimensional spherical box having a radius equal to 2Nam and the particle is randomly placed in the cell. The polyelectrolyte and the oppositely charged particle are then allowed to move (a random motion is used to move the particle). After each calculation step, the coordinates of both the particle and monomers are translated in order to replace the central monomer of the polyelectro-lyte in the middle of the box. It should be noted that the chain has the possibility of diffusing further away and leaving the particule surface during a simulation run (so the polyelectrolyte desorption process can be investigated). After relaxing the initial conformation through 106 cycles (equilibration period), chain properties are calculated and recorded every 1000 cycles. Owing to the large number of possible situations to investigate with regard to the modifications of kang, N, and Ci, the application of this model with regard to the actual processor speeds has currently limited the chain length to 100 monomer units (200 for the isolated chain).

To quantitatively characterize polyelectrolyte configurations in solution and at the surface such as the mean square radius of gyration <Rg >, the mean square end-to-end distances <R2ee> are calculated after the equilibration period [38]. Then to determine the position of the chain monomers along the coordinate normal to the surface, spherical layers around the surface are defined. The thickness of each layer, excepting the first one, is set to one monomer radius. Because of strong excluded volume effects between the monomers and the surface, the thickness of the first layer is arbitrarily increased to two monomer radii. To characterize the conformation of the adsorbed chain, the monomer fraction in tails, loops, and trains is considered. Two parameters, the particle surface coverage 9 and the adsorbed amount of polyelectrolyte r, are also calculated to characterize the particle surface. Surface coverage is defined as the fraction of the particle surface covered with the monomers that are present in the first layer:

a0N*

asurf where a0 is the projected area of one monomer, N* the number of adsorbed monomers lying in the first layer, asuf the surface of the particle, and the adsorbed amount r defined as r = 9/9max (11)

where 9max = Na0/asurf is the maximal fraction of particle surface area that can be covered by a fully adsorbed polyelectrolyte chain.