Anal. Chem. 2007, 79, 1639-1645
Quantitative Interpretation of MALDI-TOF Mass Spectra of Imperfect Carbosilane Dendrimers Alena Krupkova´,† Jan C ˇ erma´k,† Zuzana Walterova´,‡ and Jirˇı´ Horsky´*,‡
Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Rozvojova´ 2, 165 02 Prague 6, Czech Republic, and Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovsky´ Square 2, 162 06 Prague 6, Czech Republic
A general relationship is derived for the abundance of an imperfect dendrimer with a given number of missing constitutional repeating units in the two outmost layers. The relationship is used in the interpretation of the MALDI TOF mass spectrum of the second-generation carbosilane dendrimer prepared by the iterative divergent method. The model quantitatively describes the spectrum of the dendrimer and correctly predicts the MALDI TOF mass spectrum of its first-generation precursor. Thus, the use of well-resolved MALDI TOF mass spectra for assessing the purity of low-generation dendrimers with uniform end groups is substantiated for carbosilane dendrimers and to lesser extent for dendrimers in general. Dendrimers are a novel class of regularly branched, ideally monodisperse, synthetic polymers prepared by iterative stepwise procedures alternating activation and coupling steps. The buildup starts from the multifunctional core in the divergent approach; whereas the attachment of preformed dendrons to the core is the final step in the convergent approach. In both cases, constitutional repeating units are organized in layers around the core, each layer corresponding to a single generation.1 Thus, dendrimers of defined size and architecture can be prepared, whichsafter appropriate modification of peripheral functionalitiesscan exhibit strong interactions with various substrates due to the high surface concentration of active groups. All that makes dendrimers attractive for applications in catalysis,2 drug delivery,3 supramolecular chemistry,4 etc. The divergent preparation scheme is straightforward and theoretically leads to the monodisperse product because in the coupling step the so-called focal functionality of the monomer reacts only with peripheral functionalities of growing dendrimer, activated in the preceding activation step, but not with not-yetactivated remaining (i.e., peripheral) functionalities of the monomer. However, to obtain perfectly monodisperse dendrimer activation and coupling, reactions must be driven to completion * To whom the correspondence should be addressed, e-mail: horsky@ imc.cas.cz. † Institute of Chemical Process Fundamentals. ‡ Institute of Macromolecular Chemistry. (1) Bosman, A. W.; Janssen, H. M.; Meijer, E. W. Chem. Rev. 1999, 99, 16651688. (2) van Heerbeek, R.; Kamer, P. C. J.; van Leeuwen, P. W. N. M.; Reek, J. N. H. Chem. Rev. 2002, 102, 3717-3756. (3) Boas, U.; Heegaard, P. M. Chem. Soc. Rev. 2004, 33, 43-63. (4) Zeng, F.; Zimmerman, S. C. Chem. Rev. 1997, 97, 1681-1712. 10.1021/ac061783v CCC: $37.00 Published on Web 01/13/2007
© 2007 American Chemical Society
and side reactions must be prevented, which is difficult if not impossible in practice. Since the number of peripheral functionalities increases exponentially with the number of generations, any imperfection is strongly amplified.1 For example, the thirdgeneration dendrimer with a tetrafunctional core, prepared from a monomer with two peripheral functionalities, can be expected to contain almost 10% of dendrimers with a defect (missing constitutional repeating unit) in the outmost layer even if the degree of conversion were as high as 99.4% in the final activation/ coupling sequence. Under these circumstances, analytical tools for dendrimer characterization, especially for detection of structural defects, are essential. Three techniques have emerged as dominant in dendrimer characterizationssize exclusion chromatography (SEC), nuclear magnetic resonance (NMR), and MALDI-TOF mass spectrometry (MALDI-TOF MS).5 None of them is omnipotent; the techniques should be used as complementary since specific pieces of information could be obtained using each of them. The NMR, SEC, and MALDI-TOF MS triad proved useful in other areas of polymer science as well.6 NMR can provide information on details of dendrimer structure, in particular, on irregularities due to side reactions. The number of peaks in the spectrum quickly increases and peak overlapping occurs for higher generations, making the complete assignment difficult. The purity assessment is limited by experimental uncertainty of quantitative NMR. In the above given example, 10% of imperfect dendrimers leads to the 0.6% change in the integrals of relevant signals, which is well below uncertainty of NMR.5d Moreover, NMR is not a separation methodsit gives only average values for the whole sample. Thus, even if NMR could quantify the absent constitutional repeating units as 0.6% it would not differentiate between the contamination by 10% of the dendrimer with one missing unit and by 0.6% of the secondgeneration dendrimer. (5) (a) Malkoch, M.; Claesson, H.; Lowenhielm, P.; Malmstrom, E.; Hult, A. J. Polym. Sci., Polym. Chem. Ed. 2004, 42, 1758-1767. (b) Aulenta, F.; Drew, M. G. B.; Foster, A.; Hayes, W.; Rannard, S.; Thornthwaite, D. W.; Worrall, D. R.; Youngs, T. G. A. J. Org. Chem. 2005, 70, 63-78. (c) Blais, J.-C.; Turrin, C. O.; Caminade, A. M.; Majoral, J. P. Anal. Chem. 2000, 72, 50975105. (d) Shi, X. Y.; Banyai, I.; Islam, M. T.; Lesniak, W.; Davis, D. Z.; Baker, J. R.; Balogh, L. P. Polymer 2005, 46, 3022-3034. (e) Appelhans, D.; Komber, H.; Voigt, D.; Haussler, L.; Voit, B. I. Macromolecules 2000, 33, 9494-9503. (6) (a) Farcet, C.; Belleney, J.; Charleux, B.; Pirri, R. Macromolecules 2002, 35, 4912-4918. (b) Peacock, P. M.; McEwen, C. N. Anal. Chem. 2004, 76, 3417-3428.
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SEC separates macromolecules according to their hydrodynamic volume and thus can detect the dendrimers of various generations or multimers of dendrimers. The resolution of SEC, however, is not high enough to separate dendrimers of the same generation but with a different number of defects. Theoretically, the extent of defects can be estimated from the difference between experimental molecular weight and molecular weight expected for the regular dendrimer. Unfortunately, one-detector SEC is not an absolute method and requires calibration with standards of the studied polymer, which are not yet available for dendrimers. Molecular weights determined using calibration with polystyrene standards may be in significant error even for linear polymers7 and even more for dendrimers whose intrinsic viscosity, related to the hydrodynamic volume, does not obey the Mark-Houwink equation. SEC can provide correct molecular weight of dendrimers either directly if a light-scattering detector is used or through the universal calibration concept8 using either viscosity detector or off-line viscosity measurements; however, the precision of SEC methods is a limiting factor. In the above example, the presence of 10% of imperfect dendrimer with one missing constitutional repeating unit translates to 0.3% deviation of weight-average molecular weight from the theoretical value and to the polydispersity index of 1.0001. MALDI-TOF MS has a sufficiently high resolution to detect dendrimers with a specific number of various defects9 as long as they differ in molecular weight and are of sufficiently low molecular weight. Several factors are impeding acceptance of MALDI-TOF MS as a method of choice for assessment of structural defects in low-molecular-weight dendrimers. The spectrum may be confounding due to signals related to in-source fragmentation. Since such signals are usually found with dendrimers absorbing laser light,5c,10 a more serious shortcoming seems to be MALDI-TOF MS poor quantification. The response decreases with increasing molecular weight11 and can be altered by a slight change in composition, in the case of polymers even by different end groups.12 Moreover, the shape of the MALDITOF mass spectrum strongly depends on experimental conditions, not only on the ionization agent but also on the matrix used,11 and different spectra can be obtained even from different parts of the same preparation. On the other hand, it might be expected that distortion of the relative intensity of signals in MALDI-TOF MS, as compared to relative concentrations of corresponding species in the sample, is less significant for the dendrimers of the same generation, differing only by a small number of missing constitutional repeating units because the difference in both molecular weight and dendrimer surface will be small. We substantiate the idea in the present paper. First, the well-resolved MALDI-TOF mass spectra of imperfect carbosilane dendrimers of the first and second (7) Netopilı´k, M.; Kratochvı´l, P. Polymer 2003, 44, 3431-3436. (8) Bu, L.; Nonidez, W. K.; Mays, J. W.; Tan, N. B. Macromolecules 2000, 33, 4445-4452. (9) Peterson, J.; Allikmaa, V.; Subbi, J.; Pehk, T.; Lopp, M. Eur. Polym. J. 2003, 39, 33-42. (10) Baytekin, B. B.; Werner, N.; Luppertz, F.; Engeser, M.; Bruggemann, J.; Bitter, S.; Henkel, R.; Felder, T.; Schalley, C. A. Int. J. Mass Spectrom. 2006, 249, 138-148. (11) Williams, J. B.; Chapman, T. M.; Hercules, D. M. Anal. Chem. 2003, 75, 3092-3100. (12) Belu, A. M.; DeSimone, J. M.; Linton, R. W.; Lange, G. W.; Friedman, R. M. J. Am. Soc. Mass Spectrom. 1996, 7, 11-24.
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generations are presented. Using information thus obtained, we derive the relationship for the abundance of a general dendrimer with an arbitrary number of missing constitutional repeating units in the two outmost layers, assuming that the dendrimer is prepared from a perfect dendrimer (including the core as a zerogeneration dendrimer) by repeating two times activation and coupling steps. The relationship is then successfully applied to the quantitative analysis of the MALDI-TOF mass spectrum of the imperfect carbosilane dendrimer of the second generation. Parameters thus obtained predict a MALDI-TOF mass spectrum of the first-generation dendrimer precursor in agreement with experiment. EXPERIMENTAL SECTION Materials. All solvents and chemicals were obtained from Aldrich. Carbosilane dendrimers were prepared by standard iterative procedures alternating hydrosilylation with dichloro(methyl)silane catalyzed with Karstedt catalyst (platinum(0) tetramethyldivinyldisiloxane complex) and nucleophilic displacement of a chlorine with allylmagnesium bromide.13 In optimization of reaction conditions, several series of first- and second-generation dendrimers were prepared. The second-generation dendrimer with highest content of structural defects (dendrimer G2) and its precursor (dendrimer G1) were chosen for the analysis reported. MALDI-TOF Mass Spectrometry. Tetrahydrofuran solutions of the matrix (anthracene-1,8,9-triol; 20 mg/mL), dendrimer (either G1 or G2; 10 mg/mL), and ionizing agent (silver trifluoroacetate, 10 mg/mL) were mixed in the ratio 20:4:1 and deposited on the target plate (0.5-1 µL; the dried droplet method). MALDITOF MS spectra were acquired with a Biflex III mass spectrometer (Bruker Daltonics) using the positive ion, reflectron mode and delayed extraction. The spectra were summed over 500 shots with a N2 laser emitting at 337 nm. Peak integration was carried out in time domain. Only three most intensive isotopic peaks from each multiple were used in integration, and the result was converted to the value for complete isotopic distribution by a factor obtained from a simulated spectrum.14 The values of M/z are based on the external calibration and reported without internal recalibration. RESULTS AND DISCUSSION MALDI-TOF Mass Spectra of Imperfect Dendrimers. For the present purpose, the second-generation carbosilane dendrimer with the highest content of structural defects (dendrimer G2) and its first-generation precursor (dendrimer G1) were chosen from those prepared. Their MALDI-TOF MS spectra are given in Figures 1 and 2. All significant peak multiples in the spectra denoted in the figures by monoisotopic values of M/z can be assigned to Ag+ adducts of a perfect dendrimer or of the dendrimers with a certain number of constitutional repeating units missing (Table 1). The assignment is also confirmed by the observed isotopic patterns; see the inset of Figure 2 for an example. The distance between monoisotopic peaks is constant and corresponds to the molecular weight of a constitutional repeating unit and an end group (see Figure 3). Moreover, some of other multiples in Figure 1 are not related to the dendrimer (13) Zhou, L. L.; Roovers, J. Macromolecules 1993, 26, 963-968. (14) Yergey, J. A. Int. J. Mass Spectrom. Ion Phys. 1983, 52, 337-349.
Table 1. Experimental and Predicted Values of Monoisotopic Molecular Weight and Relative Signal Intensity for First and Second Generation Dendrimers (ionized with Ag+) with Various Numbers of Defects ga
db
Mec
Mpd
(I/Ig,0)ee
(I/Ig,0)pf
1 1 2 2 2 2 2
0 1 0 1 2 3 4
803.19 677.14 1811.94 1685.73 1559.70 1433.46 1307.31
803.32 677.23 1812.07 1685.99 1559.90 1433.81 1307.72
1 0.247 1 0.769 0.535 0.265 0.105
(1) 0.277 (1) g g g 0.105
a Generations. b Number of missing constitutional repeating units. Experimental values of the monoisotopic molecular weight for dendrimer adducts with Ag+. d Theoretical values of the monoisotopic molecular weight for dendrimer adducts with Ag+. e Experimental signal intensity relative to the intensity of the perfect dendrimer. f Signal intensity relative to the intensity of the perfect dendrimer calculated as described in the text. g Data used in calculation.
c
Figure 1. MALDI-TOF spectrum of the first-generation carbosilane dendrimer G1. The indicated values of m/z correspond to the monoisotopic peaks of multiples.
Figure 2. MALDI-TOF spectrum of the second-generation carbosilane dendrimer G2. The indicated values of m/z correspond to the monoisotopic peaks of multiples. Expanded spectrum of the dendrimer with two constitutional repeating units missing is given in the inset (lower curve) and compared with the simulated spectrum14 with resolution 5000 (upper curve).
G1 because they are also observed in a blank experiment without the dendrimer. The absence of any peak corresponding to the structures with chlorine means that the nucleophilic displacement proceeded to completion. The relative content of each imperfect dendrimer was determined from the spectra as an area under the corresponding multiple divided by the area for the perfect dendrimer. The values collected in Table 1 are, however, unreliable per se because of generally problematic quantification by MALDI-TOF MS. Their verification by an independent method is not available at the present time, but partial verification can be done by establishing the overall compliance of the results with the theoretical relationships for the abundance of a dendrimer with an arbitrary number of missing constitutional repeating units. In order to derive such relationships, the mechanism by which carbosilane dendrimers lose constitutional repeating units should be known; however, the
Figure 3. Structure of a perfect second-generation carbosilane dendrimer with indicated core, constitutional repeating unit and end unit, and corresponding monoisotopic and average molecular weights. Upon deletion of one constitutional repeating unit, one end group is also lost and the molecular weight of an imperfect dendrimer is decreased by 85 + 41 ) 126, which is the spacing observed in the experimental spectra (Figures 1 and 2).
exact mechanism is still disputable in the case of carbosilane dendrimers. Lorenz et al.15 considered missing constitutional repeating units to result from incomplete hydrosilylation. Allgaier et al.16 rejected this explanation, at least for their samples, because NMR spectra of the chlorosilane intermediates showed no trace of remaining alkenyl groups. Instead, these authors ascribed structural defects to the contamination of alkenyl bromide by alkanyl bromide. However, the M/z values should then be increased by 2, 4, 6, ... for dendrimers with 1, 2, 3, ... missing dendrons. No such shift was observed in our experiments, and M/z values of imperfect (15) Lorenz, K.; Mulhaupt, R.; Frey, H.; Rapp, U.; Mayer-Posner, F. J. Macromolecules 1995, 28, 6657-6661. (16) Allgaier, J.; Martin, K.; Ra¨1der, H. J.; Mu ¨ 1llen, K. Macromolecules 1999, 32, 3190-3194.
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dendrimers corresponded to all end groups being unsaturated. Moreover, no alkanyl groups in the dendrimer periphery were evident in NMR spectra (data not shown here), in which, however, small peaks due to internal double bonds appeared. Migration of the terminal double bond in an alkene is a frequent reaction in the hydrosilylation of 1-alkenes, and the internal olefins thus produced do not typically undergo hydrosilylation.17 Thus, the double bond isomerization is a third possible mechanism leading to loss of constitutional repeating units in allylbased carbosilane dendrimers. Obviously, this mechanism cannot be exclusive since missing constitutional repeating units were observed also with vinyl-based carbosilane dendrimers.16 Fortunately, the three proposed mechanisms may be merged to a general reaction scheme accounting for missing constitutional repeating units in carbosilane dendrimers. In accordance with our MALDI-TOF MS results, complete displacement of chlorine by allylmagnesium bromide is assumed in the scheme but two types of terminal groups are allowed, differing by their reactivity to dichloro(methyl)silane. The overall hydrosilylation may be incomplete. The remaining double bonds may react in subsequent hydrosilylation. Lorenz et al.15 assumed uniform nucleophilic displacement followed by incomplete hydrosilylation; Allgaier et al.16 assumed zero reactivity for alkanyl terminal groups and full hydrosilylation for alkenyls. Even though occurring during the hydrosilylation, the double bond isomerization can be operatively assigned to the nucleophilic displacement and described by the scheme, which allows for reduced reactivity of internal double bonds and not only for their total resistance to hydrosilylation. With the mechanism of constitutional repeating units loss adequately described, the theoretical relationships for the abundance of a dendrimer with an arbitrary number of missing constitutional repeating units will be derived in the following subsection. In order to make the derived equations usable for other dendrimers as well, further generalization is done. The scheme is recast in terms of an iterative divergent method of dendrimer preparation:18 hydrosilylation becomes a coupling reaction (new branching points are introduced) and nucleophilic displacement becomes an activation reaction (next coupling reaction is made possible) with the exception of the final nucleophilic displacement, which is taken as terminal group modification. Dichloro(methyl)silane, HSi(CH3)Cl2, has two peripheral functionalities (Cl) besides one focal functionality (H); however; arbitrary functionality of a constitutional repeating unit as well as of a core is assumed in the following section, where the scheme is applied to buildup of two additional layers on a perfect dendrimer. Theoretical Abundance of Imperfect Dendrimers. The general relationship for defect probability will be derived for a dendrimer prepared by a divergent method in which activation and coupling steps are alternating. The end group modification is allowed but assumed to be complete. The reactions are considered to be pseudo first order owing to excess of reagents over dendrimers. The process starts by activation of a c-functional core. Total conversion is assumed in all activation steps; however, two types of activated functionalities differing by the reactivity are (17) Stein, J.; Lewis, L. N.; Gao, Y.; Scott, R. A. J. Am. Chem. Soc. 1999, 121, 3693-3703. (18) Grayson, S. M.; Fre´chet, J. M. J. Chem. Rev. 2001, 101, 3819-3867.
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postulated. The probability that the formed activated functionality is that of a higher reactivity is Ri, where the subscript i corresponds to the generation of a dendrimer being activated. The formation probability of less reactive activated functionality is then 1 - Ri. In all the coupling steps, a monomer with f peripheral functionalities is attached by the focal functionality to an activated functionality of the dendrimer. The number of peripheral functionalities of a regular ith generation dendrimer is
ni ) cf i
(1)
The monomer attachment probability is different for more and less reactive activated functionalities of an ith generation dendrimer (denoted βi and β′i, respectively). The overall probability, γi, that a monomer is attached to a particular functionality after one activation/coupling reaction set is
γi ) Riβi + (1 - Ri)β′i
(2)
The probability, pi+1,j, that a (i + 1)-generation dendrimer with j monomeric units in the peripheral layer is formed by a activation/ coupling sequence from a regular i-generation dendrimer is then obtained in a straightforward manner as
pi+1,j )
() ni j
γij(1 - γi)ni-j
(3)
Equation 3, which was already derived by others,19 describes the distribution of (i + 1)-generation dendrimers with various number of monomeric units in the peripheral layer. The product of the terms with γi gives the probability of one particular configuration with j reacted and ni - j unreacted functionalities on an ith generation dendrimer; the prefactor gives the number of such configurations. The subsequent activation reaction can proceed only on jf newly formed nonactivated functionalities and the probability, pi+2,j,k, that after completion of the second activation/coupling sequence k monomeric units will be in the (i + 2)th layer is obtained in analogy to eq 3 as
pi+2,j,k )
() jf k
γi+1k(1 - γi+1)jfi-k
(4)
In addition, the second coupling reaction can proceed also on ni - j activated functionalities unreacted during the first coupling. The proportion of those highly reactive is distorted by the first coupling reaction. Taking this into account, the probability, δi, that a functionality unmodified by the first coupling reaction will react in the second coupling is
δi )
Ri(1 - βi)βx + (1 - Ri)(1 - β′i)β′x Ri(1 - βi) + (1 - Ri)(1 - β′i)
(5)
New values of reactivity (βx and β′x) were assigned to surviving activated functionalities for the second coupling reaction because the reasons for the difference between βi and βi+1 is not specified. If the reasons were only different steric constraints in layers i
and i + 1, then βx ) βi; on the other hand, if the reasons were only different reaction conditions of the first and second coupling reactions then βx ) βi+1. The probability, p′i+1,j,l, that a dendrimer has in the (i + 1)layer j monomeric units from the first coupling reaction and l monomeric units from the second one is again obtained in analogy to eq 3 as
( ) ni - j l
δil(1 - δi)ni-j-l
∑
Mw )
∑
(
pm MC + MRc
m)0
MC + MR
fj
pm )
fk
∑∑
pi+1,jpi+2,j,kp′i+1,j,m-j-k
(7)
j)sj k)sk
where sj is the smallest nonnegative integer greater or equal to (m - ni)/f, sk is the smallest nonnegative integer greater or equal to (m - ni), fj is a smaller number from m and ni, and fk is a smaller number from m - j and j f. Equation 7 gives the abundance of dendrimers with various numbers of monomeric units in the two outmost layers, that is with different number of defects (missing monomeric units) d ) ni - m. Although the model is based on eight microscopic probabilities (reaction degrees), the distribution is described by three aggregated probabilities corresponding to the probabilities that a terminal group reacted in a certain activation and coupling reaction pair (γi, γi+1) and that a terminal group not reacting in the first pair reacts in the second one (δi). Thus, the relationship for the abundance of a dendrimer with a given number of constitutional repeating units derived is valid not only for all preparation schemes consistent with the initial scheme but also for all other schemes that can be reduced to the above-defined three probabilities. Equation 7 completely describes defect statistics and thus can be used in calculation of various quantities related to the whole sample and that can be in some cases measured by methods other than MALDI-TOF MS. For example, multidetector SEC gives the number-average molecular weight (Mn), the weight-average molecular weight (Mw), and the polydispersity index (PI) that can be calculated using eq 7 as (19) Meijboom, R.; Hutton, A. T.; Moss, J. R. Organometallics 2003, 22, 18111815.
c+ pm(MRm + ME(cf i + mf - m)) (8)
m)0
(6)
The product pi+1,j pi+2,j,k p′i+1,j,l gives the probability that a dendrimer molecule prepared by a double repetition of an activation/coupling sequence from a regular dendrimer of ith generation contains j monomeric units attached in the first coupling, l in the second coupling in the (i + 1)th layer, and k monomeric units in the (i + 2)th layer, the total of monomeric units in the two outmost layers, m, being j + k + l. In order to get the probability, pm, that a dendrimer contains m monomeric units in the two outmost layers, the values of the product must be summed over all combinations of j, k, l that sum to m
f-1
cf i(1+f)
cf i(1+f)
p′i+1,j,l )
fi-1
Mn ) MC + MR
fi-1 f-1
fi-1 f-1
)
2
+ MRm + ME(cf i + mf - m)
cf i(1+f )
c+
∑
pm(MRm + ME(cf i + mf - m))
m)0
(9) PI ) cf i(1+f )
∑
(
(
pm MC + MRc
m)0
MC + MR
fi-1 f-1
fi-1 f-1
+ MRm + ME(cf i + mf - m)
)
∑
)
2
cf i(1+f )
c+
2
i
pm(MRm + ME(cf + mf - m))
m)0
(10) where MC, MR, and ME are molecular weights of the core unit, of the constitutional repeating unit (i.e., “monomeric unit”), and of the end group, respectively. The relationship giving the molecular weight of a perfect dendrimer20 was modified for an imperfect dendrimer with a certain number of missing constitutional repeating units and used together with definition of Mn, Mw, and PI in derivation of eqs 8-10. Application of Theoretical Relationships to MALDI-TOF Mass Spectra of Imperfect Carbosilane Dendrimers. The relationships derived in the previous section are for arbitrary values of the generation of the initial dendrimer, functionality of the core and functionality of the constitutional repeating unit. In Table 2, the relationships for abundance are rewritten for the dendrimers observed in MALDI-TOF MS spectra for samples G1 and G2sthe four-functional core is used as an initial dendrimer (i.e., zero generation) and the constitutional repeating units have two peripheral functionalities. Figure 3 gives the structure of the corresponding perfect allyl-based carbosilane dendrimer with the defined core, constitutional repeating unit and end group, and corresponding monoisotopic molecular weights to be used in calculations of monoisotopic molecular weight of imperfect dendrimers and average molecular weights to be used in calculation of Mn, Mw, and PI. Five peaks are observed in the spectrum of the secondgeneration dendrimer G2; three parameters are required for description of the statistics of missing constitutional repeating units. Assuming proportionality between the peak intensity and abundance of corresponding dendrimer, we may test consistency of the spectrum and the model. The intensities cannot be normalized by the total observed intensity to give values of abundance because additional peaks must be expected to be lost in the baseline noise. Therefore, the normalization was done by (20) Tomalia, D. A.; Naylor, A. M.; Goddard, III, V. A. Angew. Chem., Int. Ed. Engl. 1990, 29, 138-175.
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Table 2. Relationships for Abundance of First- and Second-Generation Carbosilane Dendrimers (c ) 4, f ) 2) with Different Numbers of Missing Constitutional Repeating Units (Defects) no. of defects
0 1 0 1 2 3 4
abundance of dendrimera First Generation γ04 4γ03(1 - γ0) Second Generation γ04γ18 8γ04γ17(1 - γ1) 28γ04γ16(1 - γ1)2 + 4γ03(1 - γ0)γ16δ0 56γ04γ15(1 - γ1)3 + 4γ03(1 - γ0)γ16(1 - δ0) + 24γ03(1 - γ0)γ15(1 - γ1)δ0 70γ04γ14(1 - γ1)4 + 24γ03(1 - γ0)γ15(1 - γ1) × (1 - δ0) + 60γ03(1 - γ0)γ14(1 - γ1)2δ0 + + 6γ02(1 - γ0)2γ14δ02
a Symbols: γ is an aggregated probability that a particular terminal 0 group reacted in the first activation and coupling reaction pair; γ1 is an aggregated probability that a particular terminal group reacted in the second activation coupling reaction pair; and δi is an aggregated probability that a particular terminal group not reacting in the first reaction pair reacts in the second one.
the intensity of the perfect dendrimer, and the normalized intensities of three following peak multiplets were used to calculate the model parameters. The values obtained γ0 ) 0.935, γ1 ) 0.912, and δ0 ) 0.833, give insight into the reasons for missing constitutional repeating units since a relatively high value of δ0 does not correspond to any mechanism in which unreacted groups are permanently deactivated, e.g., contamination of alkenyl bromide with alkanyl bromide, as is readily seen from eq 5. Of course, this is only evidence against such mechanisms working on an exclusive basis; they can be operative concurrently with some other mechanism that leads to δ0 >0. The principal goal of the paper, however, is not the information on reaction mechanism but on MALDI-TOF MS as a tool for characterization of dendrimers. The obtained values of γ0, γ1, and δ0 predict the normalized intensity for the fifth peak in the MALDITOF MS spectrum of the second-generation dendrimer G2 identical to that experimentally found (see Table 1). Thus, the MALDI-TOF MS spectrum of the second-generation dendrimer G2 appears to be consistent with the model of dendrimer formation. More importantly, the value of γ0 obtained from analysis of the MALDI-TOF MS spectrum of the second-generation dendrimer G2 predicts reasonably well the spectrum of its precursor, the first-generation dendrimer G1 (see Table 1). To stress the importance of this finding, the reason for analyzing a related pair of imperfect dendrimers is given. It may be argued that consistency check on one imperfect dendrimer prepared from a perfect precursor is sufficient and that the derivation of eq 7 in the preceding subsection is superfluous because the defect statistics of such a dendrimer can be described by already known eq 3 with one parameter, γ0. Such an argument is, however, fallacious. Let us assume that the proportionality between the concentration of analyte in the sample and the intensity of a corresponding MALDI-TOF MS signal depends on molecular weight and the dependence can be linearized around the molecular weight of a perfect dendrimer. If the intensities of 1644 Analytical Chemistry, Vol. 79, No. 4, February 15, 2007
the perfect dendrimer and of the dendrimer with one constitutional repeating unit missing are used to determine γ0, an incorrect value will be obtained from eq 3 in this case but the intensity of the dendrimer with two constitutional repeating units missing will be predicted reasonably well. For example, if the signal intensity of the dendrimer with one missing constitutional repeating unit is overestimated by 20% and that of the dendrimer with two missing units by 40%, the latter intensity will be predicted with a relative error less than 3%. Thus, from a MALDI-TOF MS on one imperfect dendrimer sample with constitutional repeating units missing only in the outer layer, a negligible effect of molecular weight will be falsely inferred. On the other hand, the agreement between the model developed and experimental MALDI-TOF MS spectra of two interrelated samples with significantly different molecular weights implies in a much stronger way that proportionality between the concentration of a dendrimer and the intensity of the corresponding MALDI-TOF MS signal is not affected by the change in the molecular weight of the dendrimer due to the loss of some constitutional repeating units. The conclusion is valid only for carbosilane dendrimers and experimental conditions used, but the general credibility of MALDI-TOF MS for analysis of dendrimers is also increased by this finding and will increase further as the developed methodology is applied to other dendrimer types or different experimental conditions. In doing so, some limitations may be expected when working with higher generation dendrimers or dendrimers with more than one type of end group. The problem is not the theory, which can be extended both for buildup of the third imperfect layer and for incomplete activation and end-group modification, even though additional parameters would be required. Moreover, the relationships for the buildup of two layers can be applied to the MALDITOF MS spectrum of the third-generation dendrimer if the precursor, i.e., the first-generation dendrimer, is perfect. The problem can be the quality of MALDI-TOF MS spectra, decreasing with the number of generations. This is observed with carbosilane dendrimers as well as with other types and might be expected, but the surprising thing is that the spectra are poorly resolved already at molecular weights where the linear polymers are still relatively well resolved.9,16 The reason for the worsening cannot be the polydispersity of dendrimers. PI is 1.011 for our secondgeneration dendrimer as calculated from eq 10 and the values of γ0, γ1, and δ0 determined above. A lower value PI ) 1.004 is calculated with the same γ0, γ1, and δ0 for a third-generation dendrimer (with a perfect first-generation dendrimer as a precursor). Regardless of the reasons for the spectra worsening, the quantification of the MALDI-TOF MS spectra of higher generation dendrimers is hampered as both the baseline and integration limits are obscured. The presence of various end groups makes MALDI-TOF MS spectra of dendrimers more complex and similar to the spectra of copolymers. But even if the well-resolved spectra were obtained, their compliance with theory is less probable because the effect of end groups on the relative peak intensity was observed even for linear polymers21 and may be expected to be even more (21) (a) Bellu, A. M.; DeSimone, J. M.; Linton, R. W.; Lange, G. W.; Friedman, R. M. J. Am. Soc. Mass Spectrom. 1996, 7, 11-24. (b) Puglisi, C.; Samperi, F.; Alicata, R.; Montaudo, G. Macromolecules 2002, 35, 3000-3007.
pronounced for dendrimers with a significantly higher number of end groups. CONCLUSIONS The derived relationships for the abundance of dendrimers with missing constitutional repeating units allow for description of rather complex MALDI-TOF MS spectra using a small number of parameters. The success to do so in a consistent way increases MALDI-TOF MS credibility as a tool for quantification of the content of imperfect dendrimers from well-resolved MALDI-TOF mass spectra with no evidence of fragmentation. Moreover, MALDI-TOF MS is more frequently used for purity verification rather than for full quantification because the preparative goal usually is to prepare a perfect dendrimer and verification can always be done after adding a layer. On the other hand, MALDITOF MS will be generally less reliable with samples inhomoge-
neous in both generations and end groups, which also includes dendrimers with defects other than simple deletion of constitutional repeating units. Thus, in order to establish purity of a dendrimer, MALDI-TOF MS must be combined with other methods such as NMR and various chromatography techniques. ACKNOWLEDGMENT The project was supported by Academy of Sciences of the Czech Republic (Grant AV0Z40500505) and by Ministry of Education, Youth and Sport of the Czech Republic (Grant LC06070).
Received for review September 21, 2006. Accepted December 12, 2006. AC061783V
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