Drift-Limited Regime

In this section, we consider the rate of capture of polymer chains arising solely from the second term on the right-hand side of Equation 9.7, due to the electrophoretic movement of the polyelectrolyte molecules. We call this limiting situation as the drift-dominated regime. In this limit, the steady-state flux in the number concentration is exactly the same as Equations 8.16 and 8.18, derived for small electrolyte ions, 

If the moving species is an ion from the electrolyte or an isolated counterion, its electrophoretic mobility /u. is given by the Einsteinian law. 

In the drift-dominated regime, where the first, third, and fourth terms on the right-hand side of Equation 9.7 are negligible in comparison with the electrophoretic term, the constant flux in the steady state follows from Equations 9.1, 9.2, 9.20, and 9.25 as 

If Vm is the voltage difference that drives the negatively charged polymer from the donor compartment toward the positive electrode placed in the receiver compartment, it follows from Equation 9.26 that 

Therefore, in the drift-limited regime, where the effects of the diffusion, barrier, and convective flow may be ignored, the polymer flux is linearly proportional to the polymer concentration and the applied-voltage difference and is independent of the chain length. 

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Figure 6.9 Dependence of the mean FPT ti (xq) on initial particle position for two absorbing boundaries separated by distance L (a) diffusion-limited regime and (b) drift-limited regime.

Drift-limited regime In this regime, the electrophoretic mobility of the polymer dominates over diffusion and barrier contributions. For large values of the applied voltage difference Vm, the steady-state flux is linear with the polymer concentration and Vm and is independent of the chain length... 

The rate of capture of polyelectrolyte chains by a nanopore is controlled by a combination of polymer diffusion, electrophoretic drift, convective flow, EOF, and free energy barriers associated with polymer-pore interactions. We have addressed the consequences of these contributing factors and presented explicit equations for use in interpreting experimental observations. Basically, there are the following four limiting regimes. 

Diffusion-limited regime When the diffusion of the macromolecule due to thermal motion dominates over the drift arising from any externally imposed flow fields and the barrier contributions, the steady-state flux is given by the law,... 

Overall, the capture rate falls either within the barrier-dominated regime or drift-dominated regime, and the diffusion-limited behavior is not dominant for polyelectrolyte transport under an applied voltage. 

Therefore, the two limits in the diffusion- and drift-dominated regimes for the average translocation time, for N mo, are given by... 

Several instniments have been developed for measuring kinetics at temperatures below that of liquid nitrogen [81]. Liquid helium cooled drift tubes and ion traps have been employed, but this apparatus is of limited use since most gases freeze at temperatures below about 80 K. Molecules can be maintained in the gas phase at low temperatures in a free jet expansion. The CRESU apparatus (acronym for the French translation of reaction kinetics at supersonic conditions) uses a Laval nozzle expansion to obtain temperatures of 8-160 K. The merged ion beam and molecular beam apparatus are described above. These teclmiques have provided important infonnation on reactions pertinent to interstellar-cloud chemistry as well as the temperature dependence of reactions in a regime not otherwise accessible. In particular, infonnation on ion-molecule collision rates as a ftmction of temperature has proven valuable m refining theoretical calculations. 

This regime involves forces which are so strong that the ligand undergoes a drift motion governed by (3) in the limit that the fluctuating force aN t) is negligible compared to the applied force. In this case a force of about 800 pN would lead to rupture within 500 ps. 

In the binary-electrolyte experiments carried out at large, constant cell potentials, the cell current is ohmically limited. If the conductivity of the solution is proportional to the concentration of electrolyte, the current density at a given overpotential is then proportional to Cb. Under this regime, the concentration cancels out of Eq. (2.3), and the velocity is proportional to the applied potential. For this special case, the velocity can be expressed in terms of the anion drift velocity [27, 28]. For a binary solution, this is equivalent to replacing (1 — t+) by t and i by the ohmically limited current density. 

By Equation 2.2, K depends on E at any E. In practice, a quadratic leading term means that the variation of K exceeds the measurement uncertainty and becomes noticeable fairly abruptly above some E/N threshold, as observed in experiment (2.1). At lower E/N, called the low-field limit, K may be deemed independent of E/N. Conventional IMS is usually operated in that regime, as evidenced by linearity of v with respect to E/N varied by changing the drift voltage or gas pressure... 

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