mem-brane: counterions evaporate and lead to an enthalpy-driven attraction
Christian Fleck and Hans Hennig von Gr¨unberg The possible occurrence of a phenomenon known as counterion condensation is perhaps the most prominent feature of polyelectrolyte suspensions. These chainlike macromolecules become charged, when solved in a liquid characterized by the Bjerrum lengthB
=e 2
=kT
(with e being the elementary charge, the dielectric constant of the solvent andkT the thermal energy). The counterions leaving the surface can either stay in the vicinity and thus under the influence of the charged poly-mer (bound counterions), or they can free themselves from the field of the polyelectrolyte (free counterions).
Ifd, the mean distance between two charges on the poly-electrolyte, is large compared toB, the number of free counterions in the suspension will increase in parallel with the line charge density = 1=d. However, once the line charge density becomes so high thatB
1, i.e., d B, the number of free counterions ceases to grow with and remain constant. New counterions produced by further increasing , will now become bound counterions. This behavior is reminiscent of the coexistence of saturated vapor pressure and liquid in the usual condensation process. Therefore, the phenomenon is called ”ion condensation”.
This project is concerned with the question if the reverse process is possible, that is, if something like
”counterion evaporation” or ”counterion release” can occur, and if this can lead to an extra attraction between the polyelectrolyte and the object which is responsible for the evaporation. To answer these questions we have considered a model system consisting of an infinitely long, charged, cylindrical rod that is immersed in an unbounded electrolyte and brought into the external field of an oppositely charged, planar wall.
We have solved the Poisson-Boltzmann equation to obtain the mean-field electric potential(r )in the region filled by the electrolyte solution. The inset of Fig. (1.a) shows such a potential. With(r)one can then calculate quantities such as the total grand potential energy of the system, the total entropy, the number of particles in the system, and the enthalpy. Doing this for various distancesh between the rod and the wall, one obtains the grand potential as a function of the distanceh – a
1.0 100.0
Figure 1: Change of a) particle number and b) grand potential of a system consisting of small electrolyte ions, a polyelectrolyte rod and a charged wall, as a function of the rod-wall distanceh. The numbers labeling the curves are a measure of the surface charge density of the wall. Taken from Ref. (1).
function which may be regarded as the effective wall-rod interaction potential –, but also the entropy or particle number as a function ofh.
All these quantities can now be used to analyze quanti-tatively the counter-ion release force. Fig. (1.a) shows for various different wall surface charge densities w
how the number of microions that are involved in the screening of the polyelectrolyte and the wall changes as a function of h. The counterion release is clearly visible: ions leave the system, they are ”evaporated” due to the presence of the external field of the oppositely charged wall. Returning to the language of the gas-liquid phase-transition, the electric field strength of the wall charges here plays the role of the heat which one has to supply to transfer molecules from the liquid into the gas phase. In accordance with that picture, the total fraction of evaporated ions should grow with increasing ”heat rate”, that is increasing surface charge densitywof the wall, and the four curves in the Fig. (1.a) confirm that, indeed, this is the case.
Due to this evaporation, the whole system can gain a considerable amount of enthalpy and this gain for the system manifests itself as an attractive contribution to
the effective interaction between the polyelectrolyte and the wall. Fig. (1.b) shows the grand potential as a function ofh, for four different values ofw. One can see how an additional minimum is formed on increasing
w. This additional attractive force is due to the release of counterions observed in Fig. (1.a). Ref. (1) is the first full mean-field study of this force. By analyzing the entropy and enthalpy as a function of h we could show on what mechanism this attractive force is exactly based. Contrary to what has been predicted by others, the counterion-release force is not a entropy-driven force. On the contrary: the total entropy of the system is considerably reduced due to the loss of particles, which alone would lead to repulsion. However, only parts of the total entropy are relevant for the grand-potential.
This part is the entropy of only those microions that remain in the system which is in fact the enthalpy.
This latter quantity now increases with decreasing h which is not surprising since the disappearance of ions leaves more space to the counterions remaining in the system. An increase of enthalpy means a lowering of the grand potential. Thus, the counterion release leads to an effective attraction due to a favorable change of enthalpy of the system: the counterion-release force is thus a enthalpy-driven force. This interaction, by the way, is in many respects reminiscent of the depletion interaction in a system consisting of small and large hard spheres where a gain of entropy of the whole system leads to an effective attraction between the large spheres.
That there is energy to be gained from the release of counterions is a fact long known in the theory of polyelectrolyte-ligand binding. Relatively new, however, is the incorporation of these ideas into a theory of poly-electrolyte adsorption. Our work as well as the cur-rent interest in counterion-release is triggered by a num-ber of experiments of R¨adler and coworkers on cationic lipid DNA complexation. We have chosen the parame-ters of our calculation so as to simulate the experiments of R¨adler et al., i.e., a DNA in front of a cationic lipid membrane. Only recently a paper appeared where, for the first time, counterion-release has been observed directly.
This study seems to confirm that the counterion-release force is a force that is of fundamental importance for our understanding of the adsorption behavior of DNA on op-positely charged membranes.
[1] C. Fleck and H.H. von Gr¨unberg, PRE, in press, (2001).