Ionization Mechanisms in UV-MALDI
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4. The Quantitative CPCD Rate Equation Model
Primary Ionization
Secondary Ionization
Loss Mechanisms
Example Results
4a. Primary Ionization
The concepts of the CPCD model can be cast into quantitative form,
making them much more useful for prediction and understanding of MALDI
spectra (Knochenmuss,
J. Mass
Spectrom., vol. 37, p. 867 (2002)). The unimolecular and
bimolecular (pooling) processes involved are sketched in Fig. 9.
Figure 9. Processes included in the UV-MALDI rate equation model. Group (a) includes laser-induced processes, group (b) non-radiative decay, (c) and (d) pooling. See Knochenmuss, J. Mass Spectrom., vol. 37, p. 867 (2002).
The corresponding diffrential equations are shown on this page. The bimolecular processes are scaled by D, the rate of collisions in the expanding plume. Because the plume begins at very high pressures and the external environment is vacuum (or at most 1 atm), it is can be well described as an isentropic expansion. This yields a curve like that found in Fig. 1.
Numerical integration gives the population of all states and ions as a function of time. At some long time, the plume has expanded sufficiently that the ion population no longer changes significantly. This represents the final ion yield. Becase the laser is strongly absorbed by the matrix, its intensity decreases with depth in the sample. It is therefore necessary to perform a separate calculation for multiple layers in the sample, taking into account laser attenuation (which may not follow Beer's law because of the coupled radiative processes shown in Fig. 9a). A typical result for one layer, at high laser intensity is shown in Fig. 11
Figure 11. Typical result of numerically integrated UV-MALDI differential equations. DHB matrix, 5 ns 355 nm laser pulse.
This model has proven to successfully reproduce many matrix-only MALDI phenomena. These include:
Among the more important effects for practical applications are the fluence dependence, the apparent fluence threshold, the wavelength dependence, and the spot size effect. These are all associated with readily changed parameters, and therefore deserve more attention.
Fluence vs. irradiance
MALDI has been shown to depend little on the temporal width of the
laser pulse. From femtoseconds to hundreds of nanoseconds, the
efficiency is about the same. The nature of the ions observed also does
not change. At the same time, MALDI is very sensitive to the amount of
energy per laser pulse and the area of the focused laser spot. In other
words, MALDI is fluence dependent (J/m2), but not irradiance
dependent (W/m2). This is another indication that direct
3-photon absorption is not a primary ionization pathway, because this
would be highly irradiance dependent. Instead, it shows that the lower
matrix excited states are critical to UV-MALDI. Via exciton pooling and
hopping they lead to ionization, and because they nonradiatively decay
to heat, they provide the energy for desorption/ablation. Both
ionization and heating are dependent on the lifetime of the matrix S1
electronic state. Ionization happens while it lives, heating after
it decays. The S1 lifetime, more than the laser pulse width,
is important to MALDI.
Apparent fluence threshold
A commonly observed MALDI phenomenon is the apparent fluence threshold.
At low laser fluences no signal is observed. At some threshold fluence
signal "suddenly" appears, then rises rapidly with increasing energy.
While the lack of signal at lower fluence is to some extent an artifact
of the instrumental detection efficiency and hence settings, the effect
is so pronounced that it deserves some discussion.
Because ionization involves multiple excitation, hopping and pooling steps, it is dependent on the laser fluence in a highly nonlinear manner. This alone can lead to an apparent threshold at a given detection efficiency. In addition, low fluence leads only to desorptive ion emission from the sample surface. The majority of the ions are deeper in the material and are not released. Only when ablative phase explosion becomes widespread can a larger fraction of the available ions reach the gas phase.
Wavelength dependence
UV-MALDI works better if the matrix more strongly absorbs the laser
light. By "better" we mean that the apparent fluence threshold is
lower, and the ion signal rises faster with increasing fluence. This is
also readily understood from the model. If the number of excitations
per unit volume increases (better laser absorption), the rate of
pooling and ionization rises even faster (non-linear dependence). In
addition, the same volume of material is more rapidly heated to a
higher temperature. This aids in early and complete conversion to the
gas phase. Both ionization and desorption/ablation are positively
affected by better laser absorption, reinforcing each other. These
effects are well reproduced by the model, as seen in the next Figure.
Figure 12. MALDI efficiency vs. excitation wavelength and laser pulse length for DHB matrix. The upper curve for each parameter set represents the yield from the top layer of the sample, and the lower curve the integrated yield for all emitted material.
Spot size effect
The degree of laser focussing has a distinct effect in MALDI, even when
the fluence (J/m2) is kept constant. A smaller spot gives
fewer ions, as expected simply from the area. However, the ionization
efficiency of larger spots does not scale simply with the area, but
more slowly. It is not advantageous to use large spots, from an
ionization efficiency viewpoint. The reason for these trends is
connected with the plume expansion and its relevance for ion
recombination. When the plume expands rapidly, fewer ions are lost to
recombination. The radial expansion speed of the plume is the same for
all spot sizes, given by the thermal velocity of the matrix gas.
Therefore the same change of radius represents a much larger
proportional expansion for a plume expanding from a small spot, than
from a large spot. Small spot recombination is less severe.
4b. Secondary Ionization
As noted above in section 3b
on secondary ionization, analyte ions are derived from primary matrix
ions by one or more of 3 reactions:
Once again approach to local thermal equilibrium in the dense plume allows use of Arrhenius-type rate equations. For the associated activation energies, we can take advantage of the considerable effort that has been invested in non-linear free energy relationships. In particular we can estimate the activation energy from the reaction free energy, which is much easier to obtain. This requires that we decide what kind of reaction will be dominant, because the activation energies for protonation, cationization and electron transfer may not all have the same relationship to the reaction free energy. Assuming proton transfer (typical for biomolecules) we can extend the model to include secondary reactions, without adjustable parameters.
Figure 13. Rate equations for matrix-analyte secondary reactions in the MALDI plume. The prefactor A depends on the molecular effective diameter, D, the mole fraction in the sample, F, and the mean thermal velocity, V. The activation energy parameter lambda is near 15 kJ/mol for proton transfer reactions. The volume correction is necessary because the plume is very dense and the analytes can be large compared to matrix. See Knochenmuss, Analytical Chemistry vol. 75, p. 2199 (2003).
Similar equations can be written for each analyte, and for reactions of analytes with each other. It should be noted that smaller reaction free energies (less favorable reactions) lead to lower rates and less analyte ion yield. Strongly reacting analytes can reduce the signal from less reactive ones both by more efficient depletion of matrix ions or by direct reaction between analytes. By these routes, both the matrix suppression effect and the analyte suppression effect are readily and correctly predicted by the model.
Figure 14. Comparison of calculated (left) and measured (right) MALDI spectra, with emphasis on the matrix suppression effect.
Figure 15. Comparison of calculated (left) and measured (right) MALDI spectra, with emphasis on the analyte suppression effect.
Phenomena which are well described by the model include:
The matrix and analyte suppression effects are extreme examples of a critical analytical concern: response factors in mixtures. Analysis often must go beyond molecular weight determination or a presence/absence test. If quantitative or semi-quantitative information is needed, then the change of signal vs. change of concentration in the sample must be known. The model shows that this response is a complicated, nonlinear function of composition of the MALDI sample and experimental conditions. The choice of matrix, the properties of the other analytes, the laser fluence and focussing, sample homogeneity, etc. will all play a role. At the same time, the model allows us to explore these characteristics to understand when and to what degree MALDI results can be quantitatively interpreted.
Generally, a higher matrix/analyte mole ratio leads to better relative intensities, at a cost in overall sensitivity. The sensitivity for any analyte will be better, and more reproducible if it has the most favorable secondary reaction thermodynamics. More primary matrix ions at higher laser intensity reduces selective suppression effects, giving more accurate intensity ratios. Favorable secondary reactions depend on matrix choice, the available thermodynamic information should be used, even if analyte properties can only be estimated. Similarly, cationization can be systematically planned via choice of matrix and metal salt.
4c. Loss Mechanisms
As important as any formation mechanism, loss processes play a large role
in a MALDI event. The laser pulse causes a transient excursion away
from equilibrium, one aspect of which is the formation of many separated charges.
The system will then relax back toward equilibrium, these charges will recombine to form neutrals.
At full equilibrium, the plume would be entirely neutral, no MALDI mass spectrum would be observed.
The rate of this relaxation has a large influence on the final result. If recombination
is very fast, few ions will survive the plume expansion, and MALDI sensitivity
will be low.
In the rate equation model, there is a single rate constant for recombination of positive ions with negative ions, kIO. This is then modulated by the density drop during the expansion, as are all second order processes. Since the process is initially in the condensed phase, relevant data for solid or liquid ion neutralization was sought. While no exactly corresponding data were found, neutralization of ions in the solid state by electrons trapped in their coulombic potential seemed reasonably close. The corresponding time constant of a few picoseconds gives ion yields (obviously in combination with formation rates and processes) which correspond resonably well to what is observed.
In the molecular dynamics model (next page), it is necessary to specify the recombination process in more molecular detail, a rate constant is not sufficient. This has led to investigation of what molecular properties and charge transfer mechanisms might be responsible for the observed recombination rate.
4d. Example Results
To provide some insight into the effect of key parameters, look at this page.
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