Excited State Thermal Ionization in MALDI

1. Description of the problem

Allwood, Dyer and Dreyfuss (Rapid Commun. Mass Spectrom. 11, 499, 1997) suggested that excited state thermal ionization (ESTI) could be a source of ions in MALDI:

M + hν→ M*

M* + ∆H →M⁺+ e^-

To include this possibility in numerical models, we need the rate constant for the second, ionization, step.

2. Prior Estimates of the Thermal Ionization Rate Constant

2.1 Allwood

Allwood used detailed balance and a series of approximations to arrive at:

k_ion=1.2e16 exp(∆E/kT)

where ∆E=IP(M) - E(M*)

As an example, at 1000 K and ∆E=1.1 eV (DHB), k_ion=2.7e10 /s. The prefactor is large enough for this mechanism to make a non-negligible contribution to the CPCD ion yield, under some conditions.

2.2 Alonso

Alonso and Zenobi (Phys. Chem. Chem. Phys. 18, 19574, 2016) used a very similar approach but somewhat different approximations to calculate the rate constant for:

M^-+ ∆H → M + e^-

Which is effectively the same problem as ESTI. The resulting equations must be numerically integrated, but for DHB in a MALDI simulation he obtained rates on the order of 1e8-1e9 /s. The DHB electron affinity is 0.42 eV, so ∆E for this reaction is considerably lower than for ESTI. With ∆E on the order of 1 eV, would presumably decrease by at least an order of magnitude. Such rates would be too low to contribute significantly to MALDI, compared to the CPCD pooling yields.

3. Validity of Prior Rate Estimates

3.1 Detailed balance Ansatz

Both Allwood and Alonso used detailed balance, also known as microscopic reversibility, as the basis for their approach. It requires that forward and reverse reactions are equal at equilibrium:

k_ion[X]=k_rec[X_e][e]

where k_rec is the recombination rate constant, [X_e] is the concentration of X-e^- and [e] the concentration of free electrons.

This implies:

and, as Allwood used:

However, Alonso effectively assumed K_eq=1 to write:

k_ion=k_rec

Obviously, K_eq=1 is not correct. In fact the ionization yield will not be large, maximum realistic values at moderate fluences would be on the oder of 1e-3. K_eq is then on the order of 1e-6.

The Alonso rate constants are therefore too large by a factor of at least 1e6.

K_eq is also temperature dependent, so the rate constant k_ion must also be, but this was not included in the Alonso model.

3.2 Calculation of k_rec

The reverse rate k_rec was taken by Allwood and Alonso to be:

k_rec=(ρ_Xeρ_e/ρ_X)vσ

Where ρ is the density of states (DOS) of the respective species, v the mean electron speed and σ the electron capture cross section.

The electron capture cross sections for some matrices are known experimentally, and no longer need to be estimated.

The mean electron velocity can be easily calculated in free space, but the relevant velocity (effective mass) in matrix materials is unknown. This will be ignored here.

Allwood and Alonso used different approximations for the matrix density of states (DOS). However, both assume isolated single molecules and harmonic normal modes. In reality, most ionization occurs while the sample is solid or a very dense fluid. Intermolecular degrees of freedom must be considered in the DOS, and the intramolecular modes will be somewhat different than in free molecules. And at high internal energies/temperatures, vibrations are excited to high quantum numbers where anharmonicity becomes significant. All these factors mean that the DOS approximations of Allwood and Alonso are certainly not correct, but the errors are unknown. That errors exist is also clear from the significant differences in the rates calculated by the two nominally equivalent methods.

Fortunately, we can say something about the ratio of matrix ion and neutral density of states. In the ESTI case, M and M⁺ are very similar in their vibrational degrees of freedom. Typically, a few vibrations of M⁺ will have slightly higher or lower frequencies, depending on whether the electron is removed from a bonding or antibonding orbital. The DOS ratio ρ_Xe/ρ_X will therefore be near 1. For example for 2,5 DHB at the B3LYP/6-311(d,p)++ level, the mean ratio of neutral/ion normal mode vibrational energies is 0.994+-0.004, and the ratio of the ZPEs is 1.003.

The remaining part of the DOS factor in k_rec, ρ_e , is the DOS of the emitted electron. Allwood and Alonso assumed this to be an electron in free space, and used the expression

ρ_e =8πm²v/h³

for the DOS. This result from the Drude model is incomplete. The full expression is:

ρ_e =V8πm²v/h³

where V is the volume of the system. Allwood and Alonso implicitly assumed this to be 1 m³. In MALDI, the electrons are emitted within the solid or dense fluid. The free electron model is anyway questionable, but within the model, the effective volume is clearly not unlimited. One reasonable approximation might be the mean diffusion range before capture. This is at most (100 nm)³, given the cross sections measured for MALDI matrices. The corrected electron DOS is therefore (1e-7 m)³=1e-21 smaller than that calculated by Allwood and Alonso.

ESTI electrons have low energies, around 1 eV. This is in the band gap of solid matrix, where the DOS is formally zero. (The conduction band is approximately centered on the matrix IP, around 8 eV above the valence band.) The presence of defects, impurities and ionized matrix all lead to new levels in the band gap, but the DOS of these is low and spatially localized.

Any reasonable estimate of the electron DOS in MALDI must be orders of magnitude lower than that used by Allwood and Alonso. The resulting ESTI ionization rate is therefore also orders of magnitude lower.

4. Upper limit for thermal ionization

To get an idea of the possible range for thermal ionization in MALDI, consider the case that the forward and backward ESTI rates are very fast, so that equilibrium is approached. What then is the maximum thermal ion yield during a MALDI event? This will occur when the population of the excited matrix reaches a maximum. The thermal yield is:

η=[M*]f(ΔE)

where f(ΔE) is the solution to:

At 1000 K, f(1 eV) is about 2e-3. In a CPCD simulation of DHB a fluence of 20 mJ/cm² results in a peak Sn population of about 1e-3 so η is on the order of 1e-6. The peak CPCD pooling ion yield is about 1e-2 under the same conditions.

Note that the thermal yield must decrease as the Sn population decreases after it's peak, since fast equilibrium has been assumed. But even if the backward reaction is somehow stopped at the peak of thermal ion production, yield is still much smaller than that predicted by the CPCD pooling model. If ion yield is in the 1e-6 or lower range, as some have proposed, then ESTI is a possibly a contributor, but the rate constant arguments above suggest that it is probably not.

5. Experimental data

This is not the place for a review of data relevant to thermal models of MALDI. However, it currently appears that all data available is against a thermal model. Powerful examples include wavelength/fluence data on DHB, CHCA and ClCA (Soltwisch et al), absolute yield vs fluence data for DHB and CHCA (Ni), and comparative data on DHB isomers (Kirmess et al). Results which were claimed to support a thermal model (Kim, Ni) have been shown to be quantitatively consistent with the CPCD.

6. Conclusions

There are multiple reasons to conclude that ESTI rates estimated using the methods of both Allwood and Alonso are much too high. The approximations are too unrealistic, and lead to systematic overestimation. A straightforward calculation of the maximum possible ESTI yield clearly supports this. Also, there is considerable evidence in support of nonthermal models, but little or none in support of thermal models.

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