1. Equilibrated Conformations

To investigate adsorption processes in the polyelectrolyte-particle system, a spherical charged particle (Q = + 100, op = 35.7 A) is added to the equilibrated polyelectrolyte chain (N = 100, <5m = 3.57 A, f = 1, lB = 7.14 A) so that they are close with each other. Then the polyelectrolyte-particle complex is enclosed in a spherical cell and allowed to relax. Equilibrated conformations of the complex as a function of C and N are presented in Table 3. It should be noted that the

C,[M] N |
0 |
0.01 |
0.3 |
1 |

25 |
( |
O |
f 1 | |

50 |
o |
# |
J | |

140 |
© |
# | ||

160 |
# |
# |
C | |

200 |
$ |
o |
< |

"Because of the lack of available space different scales have been used to represent the polyelectro-lyte-particle complexes.

"Because of the lack of available space different scales have been used to represent the polyelectro-lyte-particle complexes.

extended polymer chain conformations appear smaller than-their actual size as they have been reduced for reproduction (N constant = 100).

It can be clearly seen from Table 3 that no adsorption is observed when C > 1 M. Attractive surface-polymer interactions in this domain are not strong enough to overcome the entropy loss of the polymer due to its confinement near the particle. In order to determine the adsorption-desorption limit (a chain is con-

sidered as adsorbed when it is in contact with the particle during for more than 50% of the simulation time), the ionic concentration is adjusted between 0.3 M and 1 M for each chain length. The plot of the critical ionic concentration C\ (ionic strength at the adsorption-desorption limit) as a function of N is presented in Fig. 4. To overcome the entropy loss per monomer due to adsorption, it is demonstrated that stronger electrostatic attractions, with decreasing ionic concentration, are needed to adsorb short polyelectrolyte chains. C\ increases with N from 0.34 M when N = 25 to 0.4 M when N = 200. When chains longer than 100 monomers are considered, it is important to note that C\ increases slowly to reach a plateau value close to 0.4 M. The critical electrostatic energy EC associated with this limit for a single monomer in contact with the particle surface ranges from -1.18 kBT when N = 25 to -1.03 kBT when N = 200. Hence, adsorption is achieved when the attractive energy is greater than the thermal energy, i.e., 1 kBT. These results are in accordance with the picture of polymer adsorption on flat surfaces. Indeed, in dilute solutions of polydispersed polymers, long chains are found preferentially on the surface because less transla-tional entropy (per unit of mass) is lost compared to the short ones, while they gain approximately the same (total) adsorption energy even if the adsorption

energy per segment is the same. When N = 25 adsorption approaches that on a nearly planar surface. In this case, the chain can fully spread on the surface with dimensions close to its dimensions in a free solution. When chain length is increased up to N = 140, polyelectrolytes wrap around the particle to optimize the number of contacts. The conformation of the polymer is then dictated by the particle size and is subject to the highest level of deformation (the maximum is observed when N = 140). By increasing further chain length, excluded electrostatic volume prevents any additional monomer adsorption on the surface via the formation of an extented tail in solution. Due to the formation of that protruding tail in solution, the ratio <R2>ads/<Rg>free increases to 1. Complexes with two tails were not observed during our simulations according to the size ratio between the particle diameter and polyelectrolyte length.

3. Trains, Loops, and Tails

The number of monomers in trains, loops, and tails as a function of N and Ci is presented in Fig. 5a and 5b, respectively, to give insight into the structure of the interfacial region. When the chain length is increased (Fig. 5a) with C greater than 0.01 M, the total number of monomers in trains and loops increases monotically. When Ct is less than 0.01 M and N is greater than 140 monomers, the number of monomers in trains and loops does not change because any additional monomer is expelled in tails. Beyond that critical chain length N, intra-chain repulsion outweigh the attraction between the monomers and the particle to form tails. It is worth noting that the length of the tails increases linearly with N above N. When short chains are considered, monomers are mainly in trains, whereas a few are present in loops. Short chains thus have a tendency to flatten more because this lowers the energy most.

By increasing C up to 0.03 M (Fig. 5b), the electrostatic excluded volume is decreased, allowing the particle to attract more monomers. Then, by further increasing C to the critical adsorption-desorption limit C\, the number of monomers in trains decreases while the number of monomers in loops and tails increases to reach a maximum. When Ci > Cci polymer desorption is observed so that the number of monomers in trains, loops, and tails decreases rapidly.

4. Amount of Adsorbed Monomers: Surface Coverage

Further insight into the adsorption properties of polyelectrolytes is gained by examining the amount of adsorbed polymer r and the particle surface coverage 9. The influence of C and N on r (Fig. 6) clearly demonstrates that the polyelec-trolyte's capacity to be adsorbed on the particle surface decreases with the increase of Ci and N. When N > 140 r reaches a maximum value in the range 0.1 M > Ci > 0.01 M, the exact value being dependent on the polyelectrolyte contour length. By further increasing Ci, desorption takes effect and r decreases. Variations of surface coverage 9 (i.e., the total number of adsorbed monomers in the

first layer) as a function of ionic strength and for different degrees of chain polymerization are presented in Fig. 7. When N < Nc, 9 is monotically decreasing with the ionic concentration while a maximal value is achieved when Ci is close to 0.03 M and N > Nc. These results clearly demonstrate that particle surface coverage and the amount of adsorbed polymer are not simple monotonic functions of N and Ci when the polyelectrolyte is large enough to form a tail in solution.

Theoretically, collapsed monomers are those that participate in the overcharging process and are either in contact with the surface of the particle or belong to one of the tightly packed multiple layers. With simulations, this limit is subjective and often fixed arbitrarily owing to the difficulties in deciding where the adsorbed layer ends. In this section, when overcharging is discussed, the monomers are either adsorbed (N^) or in tails (Nia'), so that the number of adsorbed monomers is defined as N^ = N - Ntail.

In Fig. 8, we plot as a function of N the variation of the number of collapsed monomers N<ads in the salt-free case. Our MC simulations and the theoretical prediction of the Nguyen-Shklovskii (NS) model [19,20] are presented, and a

qualitative and quantitative good agreement is found between the two models. A numerical solution giving a first-order transition at Nc = 151 with the formation of a protuding tail composed of 19 monomers is obtained in perfect agreement with our observations.

With increasing the size of the polyelectrolyte several key results are demonstrated: (1) the chain is fully collapsed on the charged particle as long as N < Q and N ~ and the complex is undercharged; (2) when N > Q more monomer adsorbs on the particle surface than is necessary to neutralize it, and the complex is overcharged. Accumulation of monomers close to the surface continues up to N = 151; (3) beyond that critical number of monomers, a protuding tail in solution appears. The NS model predicts a first-order transition at this point followed by a small decrease in N^5, with N rapidly reaching a plateau value. Our MC simulations follow this behavior perfectly.

We now consider the overcharging issue in the presence of added salt. All calculations were performed with Ct < 0.03 M, i.e., in the adsorption domain, before the desorption process takes place. The number of collapsed monomers as a function of N and Ct is reported in Fig. 9. By increasing the screening of the electrostatic interactions, it is clearly demonstrated that the electrostatic volume of the monomers decreases, thus allowing the adsorption of a greater number of monomers on the particle surface. This result supports the NS model,

which predicts an increase of charge inversion as k-1 decreases. In addition to the increase of charge inversion with C, Fig. 9 demonstrates that the position of the first-order transition is increased with increasing polyelectrolyte chain length. In particular, when Ct = 0.01 and 0.03 M, it is worth noting that a chain composed of 200 monomers is not large enough to induce the formation of a tail.

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