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Manifestations of Charge Induced Instability in Droplets Effected by Charged Macromolecules

Phys. Rev. Lett. 109, 148301 (2012)

Ion-release processes from droplets that contain excess charge are of central importance in determining the charge-state distributions of macromolecules in electrospray ionization methods. We develop an analytical theory to describe the mechanism of contiguous extrusion of a charged macromolecule from a droplet. We find that the universal parameter determining the system behavior is the ratio of solvation energy per unit length to the square of the ion charge density per unit length. Systems with the same value of the ratio will follow the same path in the course of droplet evaporation. The analytical model is compared with molecular simulations of charged polyethylene glycol macroion in aqueous droplets, and the results are in excellent agreement.

We illustrate the mechanism of extrusion and define the parameters critical to a theoretical examination of the mechanism. The parameters of the model contributing to the energy are the solvation energy of the linear macromolecule and its charge. In the considered model we assume that the droplet has spherical shape. As the droplet shape remains spherical the surface energy is constant and the surface tension term does not enter equation \eqref{eq:ener-total}. Based on the above considerations we express the total energy of the system as
\begin{equation}
E_{\mathrm{total}} = E_{\mathrm{elec}} + (L-{\lambda}) v_{0} .
\label{eq:ener-total}
\end{equation}
where $L$ is the length of the macromolecule, $\lambda$ is the length of the extruded segment of the macromolecule and $v_0$ the solvation energy per unit length of the macromolecule. Electrostatic energy of the straight rigid segment in vacuum is set to zero and the corresponding change in the self-interaction energy upon solvation is accounted for by the solvation energy contribution.

Illustration of a system consisting of a partially extruded macromolecule (PEG) in a droplet. Model parameters are defined. The droplet radius is $R$ and the length of the extruded segment is $\lambda$. In a conducting droplet the electric charges (denoted by ``+'') are transferred to the isopotential surface of the droplet.

We evaluate the electrostatic energy of a conducting sphere and a linear charged macromolecule using macroscopic description of the constituent parts. Using the general formula for the electrostatic energy \cite[see p. 57]{electrostatics}
\begin{equation}
E_{\mathrm{elec}} = \sum \frac{1}{2} \phi'_i q_i
\label{eq:elec-def}
\end{equation}
where $q_i$ are charges and $\phi'_i$ are electrostatic potentials at the positions of the corresponding charges without that of the charge $q_i$.

Using the technique of electrostatic images the electrostatic field from charge $q$ at distance $x$ from the conducting droplet is equivalent to field created by the system of three charges $q$, $-\frac{qR}{R+x}$ at distance $\frac{R}{R+x}$ from the center of the sphere and $\frac{qR}{R+x}$ at the center of the sphere.

Using equation \eqref{eq:elec-def} we write the contribution to the electrostatic energy from charges in the droplet (which are distributed on the surface) as
\begin{equation}
E_{\mathrm{elec}} (Q_1) = \frac{1}{2}\frac{1}{4\pi\epsilon_0 R}
\left( Q_1 + Q_c \right) Q_1
\label{eq:ee-surf}
\end{equation}
where $\gamma$ is the charge per unit length of the macromolecule, $Q_1 = \gamma (L - \lambda)$ is the charge inside the droplet and $Q_c$ is the total induced image charge in the center of the droplet given by
\begin{equation}
Q_c = \int\limits_0^{\lambda} \frac{R \gamma d x}{R+x} .
\label{eq:elec-charge-induced}
\end{equation}
Note, that there is no contribution to the electrostatic energy from the pairs of charges $(q, -\frac{qR}{R+x})$ as they compensate each other exactly on the surface of the sphere.

Contribution to the electrostatic energy from the charges on the extruded part of the macromolecule is
\begin{equation}
E_{\mathrm{elec}} (Q_2)
= \frac{1}{2}\frac{1}{4\pi\epsilon_0}
\left[ \left( Q_1 + Q_c \right) \int\limits_0^{\lambda} \frac{\gamma
d x}{R+x} - \int\limits_0^{\lambda} \int\limits_0^{\lambda}
\frac{\gamma dx}{ R + x - \frac{R^2}{R+y} } \frac{R \gamma dy}{ R
+ y } \right] .
\label{eq:ee-macr}
\end{equation}

Adding energies given by equations \eqref{eq:ee-surf} and \eqref{eq:ee-macr} and after some algebra we arrive at
\begin{equation}
E_{\mathrm{elec}} = \frac{1}{2}\frac{1}{4\pi\epsilon_0} \biggl[
\frac{1}{R} \left( Q_1 + Q_c \right)^2 - \gamma^2 R
\int\limits_0^{\lambda} \int\limits_0^{\lambda} \frac{dx
dy}{(R+x)(R+y)-R^2} \biggr] .
\label{eq:ee-total}
\end{equation}

We analyze stability of the system given by equation \eqref{eq:ee-total}. The central property is the location of the minima of the energy as a function of macromolecule extrusion $\lambda$. The locations of the minima are given by the solutions of the following equations
\begin{equation}
0 = - \frac{\partial E_{\mathrm{total}} } {\partial \lambda} = v_{0}
+ \frac{1}{4\pi\epsilon_0}
\biggl[ \frac{Q_1 + Q_c} {R} \frac{\gamma \lambda}{R+{\lambda}}
+ \gamma^2 R \int\limits_0^{\lambda}
\frac{dx}{(R+x)(R+{\lambda})-R^2} \biggr]
\label{eq:minima}
\end{equation}

Equation \eqref{eq:minima} can be explicitly evaluated and recast in the following form
\begin{equation}
-\frac{4\pi\epsilon v_0}{\gamma^2} = \biggl[ \frac{L-\lambda}{R} +
\ln{\frac{R+\lambda}{R}} \biggr] \frac{\lambda}{R+{\lambda}} +
\frac{R}{R+{\lambda}} \ln{\frac{2R+\lambda}{R}},
\label{eq:min-simpl}
\end{equation}
with dimensionless parameters on both sides of the equation. The universal parameter $B_{\mathrm{ex}}$
\begin{equation}
B_{\mathrm{ex}} = \frac{4\pi\epsilon v_0}{\gamma^2} =
\biggl[L v_0\biggr] \cdot \biggl[ \frac{Q^2}{4\pi\epsilon L} \biggr]^{-1}
\label{eq:univ}
\end{equation}
determines the position of the minima for a specific system. $B_{\mathrm{ex}}$ is the ratio of the total solvation energy over a measure of electrostatic energy of the macromolecule and could have been obtained from dimensional analysis as the only combination of two characteristic quantities of the system. The systems with the same ratios of solvation energy to the square of the charge density will follow the same path in the course of droplet evaporation on the $(\lambda/R, L/R)$ diagram.

Regions of the phase diagram in the system of $\xi = \lambda/L$ and $L/R$ coordinates. Contiguous lines correspond to constant values of the interaction parameter $B_{\mathrm{ex}}$. The gray region in all subplots indicates location of the restricted domain (see details in the text). The plot shows an overall view of the minima of the droplet energy. Representative snapshots of simulations of charged PEG in water droplets that correspond to the various regions of the phase diagram are also shown.

Following the approach used to describe gas-liquid boundary lines of the van der Waals equation of state we solved the system of equations $\bigl\{ \frac{\partial E_{\mathrm{total}} } {\partial \lambda} = 0 \wedge \frac{\partial^2 E_{\mathrm{total}} } {\partial \lambda^2} = 0 \bigr\}$ and determined that the allowed region of parameters lie on the r.h.s. of the dashed curve shown in \figref{fig:phas-diag} and given by
\begin{equation}
\frac{L}{R} = \frac{\lambda}{R} + \frac{\lambda^2}{R^2} -
\ln\frac{R+\lambda}{2R+\lambda} - \frac{R+\lambda}{2R+\lambda}.
\label{eq:allowed}
\end{equation}

Equations \eqref{eq:minima} and \eqref{eq:allowed} are conveniently presented in the ($\xi = \lambda/L$, $L/R$) system of coordinates. The system of equations were solved numerically and the results of calculations are presented in \figref{fig:phas-diag}. On the phase diagram (\figref{fig:phas-diag}) the dashed line delineates the region of $(\xi, L/R)$ values with the solutions corresponding to the maxima of the total energy (equation \ref{eq:ener-total}). In this region the fully solvated chain (corresponding to $\xi = 0$) and an extruded state lying on the boundary of the allowed region are in dynamic equilibrium as illustrated in \figref{fig:phas-diag}a lower insert. On the phase diagram we identified two distinct regions that correspond to different extrusion mechanisms.

@article{consta2012manifestations,
  title = {Manifestations of Charge Induced Instability in Droplets Effected by Charged Macromolecules},
  author = {Consta, Styliani and Malevanets, Anatoly},
  journal = {Phys. Rev. Lett.},
  volume = {109},
  issue = {14},
  pages = {148301},
  numpages = {5},
  year = {2012},
  month = {Oct},
  doi = {10.1103/PhysRevLett.109.148301},
  url = {http://link.aps.org/doi/10.1103/PhysRevLett.109.148301},
  publisher = {American Physical Society}
}