Possible Magnetic Resonance Signal Due to the Movement of Counterions around a Polyelectrolyte with Rotational Symmetry

Experimental, theoretical and computational studies revealed that the characteristic time scales involved in counterion dynamics in polyelectrolytes systems might span several orders of magnitude ranging from subnanosecond times to time scales corresponding to acoustic-like phonon mode frequencies, with an structural organization of counterions in charge density waves (CDWs). These facts raise the possibility of observing Magnetic Resonance (MR) signals due to the movement of counterions in polyelectrolytes. In case that this signal is detected in macroions or other biological systems, like micelles, vesicles, organeles, etc. with rotational symmetry, this method opens a new tool to measure with precission the counterions velocity.


Introduction
Polyelectrolytes are ionizing macromolecules.An important property of polyelectrolyte molecules is the formation of electric double layers surrounding the polymer chains.Most of the biological macromolecules under physiological conditions are polyelectrolytes in solution and their biological activity depends on their physico-chemical properties.Depending on the strength of the electrostatic interactions, it has been found [1] that distinct "phases" of counterions can be formed, i.e., a "condensed" layer of mobile oppositely charged counterions [2] [3] and "diffuse" phase consisting of loosely bound counterions to the considered macroion.The last phase of collective motion of the more mobile ions can be involved in the formation of charge density waves.
Experimental [4]- [7], theoretical [8]- [14] and computational [15]- [17] studies reveal that the characteristic time scales involved in counterion dynamics in polyelectrolyte systems may span several orders of magnitude ranging from subnanosecond times to time scales corresponding to acoustic frequencies.The counterions exhibit an acoustic-like phonon mode that suggests the existence of a correlated phase.At small length scales within the domains, counterions exhibit liquid-like correlations and dynamics, and they are organized into counterion charge density waves (CDWs) [18].The measured speed of sound is of the order of 2000 m/s.
We believe that these CDWs exist also on the surface of polyelectrolytes with rotational symmetry generating a circular current loop, which produces a magnetic field B and the corresponding magnetic moment µ at the center of the macroion.This magnetic moment µ is oriented in an external magnetic field o B producing a magnetic resonance signal under the aplication of a certain frequency ν .

Magnetic Field on the Axis of a Circular Current Loop
Consider a circular loop of wire of radius R located in the xy plane and carrying a steady current I , as shown in Figure 1.The magnetic field at an axial point P a distance z from the center of the loop is given by [19]: where o µ is the permeability of free space, ( ) is the magnetic moment associated with the current loop, where o z is a unit vector in the z direction.The magnetic field at the center of the loop, we set 0 z = in Equation (1).At this special point, this gives 1 ,

(
) In case that the current loop is produced by charged particles in movement, with electric charge q and velocity v .The magnetic moment is given by, ( ) where we have used 2π , with v R ω = , with τ the period and , v ω the angular and linear velocity of the counterions respectively.

Magnetic Resonance
A magnetic moment in a external magnetic field o B adquires an energy E given by the following escalar product [20], ( ) We observe that the highest and lowest energy is when π θ = and 0, θ = this means that when z µ and z B are in opossite directions the energy is maximum and in the same directions is minimum.This last state is the most natural tendency of the magnetic dipole, paralell to the field.The difference in energy between this two states is given, Reemplacing z µ given by Equation (3) in Equation ( 5), considering As an example consider the circular movement of counterions on the surface of a polyelectrolyte, used in nanomedicine with rotational symmetry, as shown in Figure 2, this does not mean that this effect can be visualize necessarily in this molecule.These ions with charge q and velocity v produce a magnetic field B and a magnetic moment µ perpendicular to the plane of the macroion in accordance to Equation (2) and Equation ( 3).If we put this magnetic moment in an external magnetic field o B , its orientation will no longer be random.The small magnetic moment may spontaneously "flip" from the most favorable orientation, the lowenergy state and the less favorable orientation the high-energy state and visceversa.The energy required to induce flipping and obtain a Magnetic Resonance (MR) signal, given by Equation ( 6), is shown in Figure 3 to depend on the strength of the magnetic field o B in which the macroion containing the magnetic moment is placed.The input radiation energy in order to accomplish the transition given by Equation ( 6) is given by Planck's law Using Equation ( 6), we obtain for the resonance frequency, where h is Planck's constant ( )

Conclusions
The distribution of molecular magnets in the different energy states is given by the Boltzmann equation, where upper N and lower N represent the population of molecular magnetic moments in upper and lower energy states, respectively.
is Boltzmann constant and T is the absolute temperature (K).To give some idea of the consequences of increasing magnetic field on the population of molecular magnets states, the distribution of a small number (about two million, non real2 ) of macroions magnets, calculated from Equation (9), is shown in Figure 3.
Such a small population difference presents a significant sensitivity problem for MR because only the difference in population is detected; the others effectively cancel one another.As seen from Equation ( 8) and Equation ( 9), the use of stronger magnetic fields will increase the population ratio, and consequently the sensitivity.In Table 1, we observe that the resonance energy, varies between ( )  2 are reported NMR energy and frequency data of 3 nuclei.Compared with Table 1 for the counterions, we observe that the NMR values are one to three digits lower.
In case that this signal is detected in macroions or other biological systems, like micelles, vesicles, organeles, etc. with rotational symmetry, this method opens a new tool to measure with precission the counterions velocity.The magnetic moment for phonons counterions with velocities of the order of 2500 m/s and for a macroion of 10 nm radius with

Figure 1 .
Figure 1.Magnetic field on the axis of a circular current loop. 1 The SI unit of magnetic field is the tesla (T): [ ]

Figure 3 .
Figure 3. Dependence on magnetic field strength B o of ΔE and the relative populations of the energy levels for counterions with z = 2 and velocity V = 2000 m/s and a macroion with R = 10 nm.

Table 1 .
Results shown in Figure 3.