Stochastic dynamics mass spectrometric structural analysis of poly(methyl methacrylate)

: Polymers such as poly (methyl methacrylate), comprising of all-carbon backbones are regarded as ubiquitous functional materials mainly due to their low cost, unique physical properties, and impressive robustness. The latter properties are results from relative chemical inertness of the polymer backbone carbon–carbon bonds. The highlighted stability crucially challenges not only innovations to plastic recycling polymer materials, but also developments of reliable analytical protocols for determining both quantitatively and structurally polymers. Despite, the fact that there are numerous applications of mass spectrometric methods to polymer science, the analysis involves chiefly annotation of monomers of polymers and additives, owing to the fact that the additives are low molecular weight analytes. The exact structural determining of end-chain groups, particularly highlighting chemically substituted end carbon–carbon bond represents significant challenge even utilizing the superior features of the analytical mass spectrometric instrumentation. This study, in the latter context, illustrates innovative stochastic dynamics approach and model equations capable of not only determining unambiguously substituted carbon–carbon end chain of the entitled polymer but also to predict mass spectra of its derivatives: thus, extending crucially the applicability of the method to many fields of the fundamental scientific and industrial research. The latter claim is argued still at the beginning of the study with the achieved method performances showing |r|=0.9999 8 . The study utilizes experimental and theoretical mass spectrometric data on high resolution electrospray ionization mass spectrometry ; high accuracy quantum chemical static methods, molecular dynamics ; and chemometrics .


Introduction
The soft-ionization mass spectrometry has become an important analytical tool in polymer science [1,2], analogously to its application to omics-methods for purposes of biology and medicine [3].The field of polymeromics deals with determining polymer structure via MS approaches [4].Broadly utilised ionization tools, amongst others, are MALDI-, atmospheric pressure chemical ionization, direct analysis in real time, ion mobility, and ESI-MS, respectively [3,5].
Although, superior method performances of analytical mass spectrometry [6], due to complexity of polymers there have been highlighted limitations to many MS applications to polymer analysis, such as: (i) Polymers should sufficiently ionize, thus forming stable species in gas-phase; (ii) the results, so far, have shown a lack of specific information regarding functional groups of polymers which can be obtained reliable by means of MS methods, in addition to scarce data on primary and higher-order macromolecular structures; and (iii) There is obtained mixed or blended data on molecular properties of polymers, due to difference in experimental ionization MS conditions and detection, respectively, identification efficiencies of monomeric constituents of macromolecules, and lack specificity [3,7].Therefore, MS methods are mainly used to determine molecular weight of polymers.Since, MS methodological instrumental newcomers have resulted to increase in mass range detection, there are examined polymers with high molecular mass.It is annotated monomers of polymers; end-chain groups; and additives [4,8].The analysis of additives determines LMWs.Advantages of MS methods for detecting polymers semi-quantitatively have been discussed [4].ESI-MS tends to produce multiply charged polymer species [9].Mass spectrometry is mainly used to concern data on synthesis of polymers; polycondensation reactions; polymerization process of ring opening via backbone carbon-radical reactions; conversion of carbon-radical polymerization; chain extension reaction; as well as to obtain physico-chemical properties of synthetic macromolecular materials such as their solubility; thermophysical; miscibility; and stereo-complexes, in addition to their degradation reactions [2,10].Real time polymerization monitoring by MS has been detailed [11].
However, properties of polymers are determined by their molecular topology and composition; side and degree of molecular functionalization as well as shape of distribution of molecular masses [12].
Therefore, methods capable of determining 3D molecular and electronic structures of polymers as well as their molecular architecture [13] crucially contribute to develop fields of polymer science.In-depth understanding of structure-property relation is of primary importance for designing and synthesising new multifunctional polymers.Owing to superior method performances [6] MS instrumentation is overall often used to study polymers and their chemical processes [10].So far, effort has been concentrated on MALDI-and ESI-MS analyses of chemical reactivity, molecular structure, and properties of polymers.
However, they show the following reliability of determining of polymers: r 2 =0.9980-0.9946.The capability of chromatography as analytical method of producing excellent performances has been proven (r 2 =0.999) [15].The value r 2 =0.998 [14] indicates that, due to chemical properties of polymers and complexity of their molecular structures and chemical reactivity, their both quantitative and 3D structural analyses do not represent trivial tasks even employing highly precise analytical instrumentation such as chromatography and mass spectrometry, particularly, highlighting ESI-and Fourier transform ion cyclotron resonance MS methods.Mass spectrometry is involved into polymer research, due to its capability of determining structurally linear, branched, and cyclic analytes, in addition to distinguish among types of oligomers and additives.
Moreover, it yields to both quantitative and 3D structural information about polymers in mixture instead of average analytical information about analye structure and properties.MS methods determine finest structures and quantity of analytes.They are required as assurance of efficiency, safety, and quality in polymers [10].
However, MS based omics-approaches provide relative quantitative analysis (r 2 =0.99) [16].The value does not reflect the actual instrumental features of analytical mass spectrometry [6], due to complexity of samples, matrix effect, and methods for data-processing of measurands, etc.
In overcoming the problem there is developed innovative methodology for exact data-processing of MS measurands via model equations ( 1) and (2) [17][18][19][20][21][22][23][24][25].They are applicable to quantitative and 3D structural analysis of molecules via mass spectrometry.Since, the analytical information is exact, the equations serve high standards of mass spectrometry, which is regarded as a gold standard of the analytical practice.
Function D " SD =f(D QC ) yields to |r|=0.9999 4 examining LMW enantiomers [21].Parameter D QC of equation ( 3) accounts for energy value of unique 3D molecular structure, electronic charge distribution, and isotopologies of molecules [25].Chemometrics of relation D " SD =f(D QC ) details on exact analyte 3D molecular structure.
Equation ( 4) is written on the base on equation (2) [20].It is valid to any two sets of measurands (l and m) of intensity data on q th analyte product ion produced, due to CID-MS n reaction (n>1.)Correlation coefficient |r l,m | is obtained assessing intensity data on two sets of measurands.The reliability of equation (4) shows |r|=0.99999 examining ESI-CID-MS 2 to MS 7 spectra of labetalol depending on CE and syringe infused volume.
Total average intensity (I TOT ) values of peaks according to known quantitative methods and parameters D i SD and D " SD of equations ( 1), ( 2) and ( 4) are connected via formula (5).The A D and A I parameters are obtained via SineSqr approximation of relations D " SD =f(CE) and I TOT =f(CE).
Despite, excellent-to-exact performances of formulas (4) and ( 5), so far, they are tested on few LMW systems.The same is true for equation (6) which is derived from formula (2).It predicts MS spectra of analytes using approximations <I 2 > > <I> 2 and D " SD D QC .The I SD Theor denotes approximated theoretical intensity data on peak of q th product ion depending on certain conditions of MS measurements.It excludes from assessing fluctuations of measurands.

(
) The model is simple and capable of distinguishing between positional stereoisomers when there is significant difference in intensity values of MS peaks.
Nevertheless, formula (6) does not account for fluctuations of intensity data on MS species.It is incapable of determining subtle electronic effects of tautomers and enantioners.
However, its application avoids some challenges that could occur calculating MS spectra of such product ions, theoretically.The application of equation ( 6) to predict MS spectra of LMWs, so far, has resulted to |r|=0.9992 2 -0.99 correlating data on I Theor SD and experimental I TOT av data on metabolites in plants and alanylcontaining oligopeptides [23,26].
Equation ( 6) is capable of distinguishing among cis/trans and positional isomers of species; furthermore, reliably from perspective of chemometrics.
In this light, first in the literature, the current study tests formulas ( 1)-( 6) on synthetic polymers examining poly(methyl methacrylate) and its oligomers.In order to, convince persuasively the reader of reliability of analytical structural data on polymers, it summarizes statistically representative data on macromolecules showing mass peaks spaced by Δ(m/z)=|100| such as (co)polymers of 3-hydroxylvalerate, in addition to derivatives of α-4-pentenyl-ω-(p-vinylbenzyl) polystyrene, showing Δ(m/z)=104, as well.Polymers of former type are based on poly(ε-caprolactone).They are widely used in pharmacy and medical research, due to their excellent biodegradability, mechanical properties, excellent hydrophobicity, and biocompatibility with biopolymers [27].
The polymers science is closely connected with field of environmental research, as well.Environmental pollution and management of solid-wastes also deal with growing production of plastics.Tasks involve not only analysis of environmental pollution of polymers, but also elaboration of so-called green biodegradable polymers.
Therefore, due to need of elaboration of such functional materials capable of replacing plastics, there is increasing interest in examining polyhydroxyalkanoates such as poly(3-hydroxybutyrate-co-3-hydroxyvalerate) copolymers for industrial production of new biodegradable plastics [28].They are regarded as most prominent derivatives for petroleum-based replacement of plastics [29].
One particular aspect of molecular structure of polymers has received significant attention.The industrial scale manufacturing of synthetic polymers is associated mainly with their low cost of production and robust physical properties [31].The largest scale of commodity plastics is synthesized via C-C chain growth of alkene monomers such as poly(meth)acrylates, poly-styrene, and polyolefins.The exceptional stability of synthetic polymers, regarding erosion processes is a result from inertness of C-C chain chemical bonds comprising molecular backbone of polymers.
However, the significant stability of C-C bond challenges innovations regarding production of closed-loop recycling plastics, treating plastic waste [32].Industrial scale methods for recycling polymers are based in thermomechanical approaches; thus, producing lower-quality of materials, exhibiting reduced mechanical properties.In the latter context, a particularly promising approach involves chemical recycling plastics [32,33].It utilizes chemical modeling and substitution of C-C bond of synthetic polymers for producing new re-polymerized materials from plastic waste, having variety of tunable and desired mechanical properties.
Due to, these reasons a comprehensive understanding of relationship between molecular structure of polymers and their properties together with chemical reactivity of substituted C-C bond as well as mechanistic aspects of backbone carbon-radical reactions contribute crucially to develop effective methods for chemical recycling of plastics.
Despite, the fact that the approach is particularly attractive for both fundamental scientific and industrial scale research and application there is significantly challenged the elaboration of such methods, due to exceptional stability of C-C bond and difficulty to revert polymer backbone chain to its monomer units.Common polymer degradation reaction is radical formation on molecular chain backbone.Further, C-C bond breaking via β-scission could produce degradable polymer materials despite all-carbon molecular backbones.Backbone carbon-radicals of polymers can lead to radical quench; thus, producing branched polymer structure or coupling reactions, as well.Elaboration of chemically substituted end-chain polymers has emerged as an appealing tool for producing chemically recycling plastics of all-carbon backbone polymers.For instance, poly(methyl methacrylate) is such all-carbon backbone compound produced via carbon chain-growth reaction of polymerization.The annual production of PMMA is higher than 4 million tons with expected increasing in production up to 6 million tons by 2027 [32].The governing factors determining industrial scale manufacturing of PMMA is its low density comparing with glass materials and high mechanical strength; strong toughness; and excellent resistance toward weather, and good chemical stability [32].
However, currently at about 10% of PMMA is only annually recycled.The research effort on developing of PMMA based chemically recycling materials has shown a dramatic acceleration of depolymerization reactions of PMMA depending on type of chain-end C-X chemical bond.Furthermore, microphase-separated regions of polymer material can be selectively depolymerized aiming at achieving nanopatterned materials.
End-to-end depolymerizable polymers gain significant attention, due to many other potential applications, rather than only to develop new recycling materials [34,35].In the latter case, synthesis of 3D interconnected polymer carbon-network throughout polymer matrix is a prospective approach to elaborate nanocomposites, such as carbon nanotubes.They produce core-shell structured polymeric materials of poly(methylmethacrylate) microspheres [35].
To conclude this section, let I stress what the study is discussed.Its major highlights are that innovative stochastic dynamics model equation ( 2) and its developed derivative (2023) equation ( 6) are innovative working tools for exact determining of molecular structures of synthetic polymers such as the entitled poly(methylmethacrylate) which reliable testing clearly indicates that the formulas are applicable to deal with cases of analytes where we lack detail molecular structural information about the analytes.Since, the experimental and theoretical designs and tests are based on a set of models of end-chain functional groups of polymers, the application of equation ( 2) allows to determine exactly the type of the chemically substituted C-C bond of the polymer chain; furthermore, detailing on 3D molecular geometry and electronic structures of the analyte.As we have seen in the introductory section, the mass spectrometric analysis of polymers is routinely used to determine molecular mass of analyte and LMW additives to polymers.As we shall see in the following sections that is not what formulas ( 2) and ( 6) provide to do examining polymers.They intend to provide 3D structural data on substituted C-C backbone chain of the macromolecule: thus, extending crucially the capability of the method to determine structurally not only homo-polymer and oligomer materials, but also their C-C chemically substituted end-chains, including composite materials.The following sections also shall illustrate the standard requirements that the analytical mass spectrometric analysis must satisfy if the experiment is designed for the high standards of the analytical practice valid to many fields of the fundamental science and industry ⎯ some of them briefly sketched above ⎯ which deal with analysis and application to both polymer materials and chemical analysis.The study maintains that equation ( 6) is capable of reliable predicting mass spectra of chemically substituted poly(methylmethacrylate), as well.

Materials and methods
The study uses mass spectra of polymers of Thermo Fisher Inc. database of Xcalibur 2.0.7 software (Thermo Fischer Scientific Inc.) (Consider raw-files msnboowser.raw and timap3.raw.)Experimental MS data are compared with standard sample of PMMA available, herein [36].
Poly(methyl methacrylate) standard (beads) having average molecular weight 35000 has been obtained from Thermo Fisher Chemicals (Thermo Fischer Scientific Inc.)The samples have been dissolved in solvent mixture chloroform/methanol (8:2 v/v) at a concentration of 0.5 mg.(mL -1 ) and introduced into ESI-interface at a flow rate of 58.20 mL.min -1 (see detail on Table A1.)The injection volume has been 10 µL.The potential between needle and ESI chamber has been set to 4.5 kV.The capillary temperature has been optimized up to a limit of 199.90°C.Mass spectra have been acquired over range of values m/z 300-2000 and 500-2000 in positive ion mode.ESI-MS/MS experiments have been performed via Finnigan LCQ ion trap mass spectrometer with helium or argon used as both damping and collision gases.The number-average molecular weights of oligomers and polymers have been estimated via gel permeation chromatography by the manufacturer and are 2400 and 1.08.Solutions of the analyte and alkali metal salts have been mixed in a ratio of 100:1 (v/v) prior to add 20 μL to ESI-nanospray needle.
The motivation behind employment in independent mass spectrometric detectors is increased resolution as instrumental features over linear and quadrupole ion trap detectors; thus, allowing for unambiguous determining of product peaks to molecular structures of chemicals.

Theory/Computations
GAUSSIAN 98, 09; Dalton2011 and Gamess-US [37][38][39][40] program packages were used.Ab initio and DFT molecular optimization were performed by B3PW91 and ωB97X-D methods.The Truhlar's functional M06-2X was utilized.The algorithm by Bernys determines GSs.Potential energy surface stationary points were obtained via harmonic vibrational analysis.Minima of energy are confirmed when there is a lack of imaginary frequencies of second-derivative matrix.Basis set cc-pVDZ by Dunning, 6-31++G(2d,2p) and quasirelativistic effective core pseudo potentials from Stuttgart-Dresden(-Bonn) (SDD, SDDAll, [http://www.cup.unimuenchen.de/oc/zipse/los-alamos-national-laboratory-lanl-ecps.html] were utilized.The ZPE and vibrational contributions have been accounted for up to a magnitude value of 0.3 eV.Species in solution were studied by explicit super molecule and mixed approach of micro hydration by PCM.The ionic strengths in solution was accounted for, using integral-equation-formalism polarizable continuum model.Merz-Kollman atomic radii and heavy atoms UFF topological models were used.The pH effect was evaluated computing properties in neutral and cationic forms.MD computations were performed by ab initio BOMD was carried out at M062X functional and SDD or cc-pvDZ basis sets, as well as, without to consider periodic boundary condition.The trajectories were integrated using Hessian-based predictor-corrector approach with Hessian updating for each step on BO-potential energy surface.The step sizes were 0.3 and 0.25 amu 1/2 Bohr.The trajectory analysis stops when: (a) Centres of mass of a dissociating fragment are different at 15 Bohr, or (b) when number of steps exceed given to as input parameter maximal number of points.The total energy was conserved during computations at least 0.1 kcal.mol - 1 .The computations were performed via fixed trajectory time speed (t=0.025fs) starting from initial velocities.The velocity Verlet and Bulirsch-Stoer integration approaches was used.The Allinger's MM2 force field was utilized [41,42].The low order torsion terms are accounted for higher priority rather than van der Waals interactions.The method's accuracy is 1.5 kJ.mol -1 of diamante or 5.71.10 -4 a.u.
(Consider results from RTs 3.6, 6.10 and 6.91 mins (Figure 6, and Table A2.))Correlation between CPAs and total average intensity data on MS peaks when quantifying analytes mass spectrometrically shows excellent |r|-parameter, while relation D " SD =f(CPA) is characterized by exact |r|-value examining LMWs such as steroids [15].The results assume a random mixture of n-mers of the same type (below.)Linear correlation between MS and chromatographic data assessing RTs values of homo-and PMMA (co)polymers has been shown [49].Thus, 100 % MMA homopolymer is obtained at RT=10.18 mins looking at product MS ions within m/z 1200-2000.MS peaks belonging to species with RTs=4-6 mins are assigned to co-polymers, exhibiting different mole fractions of MMA [49].The ESI-MS/MS spectra of (co)polymerization reactions of poly(δ-valerolacton) also reveal Δ(m/z)=100 of difference in major MS peaks of n-mers [52].There are product ions at m/z 785.5, 885.5, 985.5, 1085.6,1185.6,etc.The MS pattern of poly(3-hydroxybutyrate-co-3hydroxyhexanoate) copolymer is characterized by Δ(m/z)=114 and 86.The Δ(m/z) = 114 value of oligomers exhibits Na + -adducts of esters of polycaprolactone diol.Product MS ions of PMMA end-group are detailed on [50].The cationization effect of PMMA is associated with ions at m/z 284.5, 419.5, 441.5, 650.5, 859.6, 1068.6, and 1277.6,respectively.
Mass spectrometric data on PSs depending on end-chain groups have been comprehensively discussed [56,59].The MS cleavage pathways depend on type of metal ion of charged species (M + ).When, there is M + =Ag + , then cleavage of internal C-C bonds of oligomer occurs [57,58].The same is true for bulk end-chain groups of polymers such as -CH 2 CH 2 OH 17 and -Si(CH 3 ) 2 CH 2 CH 2 (CF 2 ) 5 CF 3 ) ones.
6-Hydroxyhexanoic acid MS product ions [61] are also considered theoretically, herein.Figures A8 and A9 detail on additional fragmentation species; thus, allowing for determining end-chain group via statistical assessment of relation D ''  SD =f(D QC ).Approach to determine end-chain group of PMMA assessing total intensity data on characteristic MS product ions has been elaborated [2,50].
Owing to the fact that there is difference in experimental conditions of MS measurements of samples looking at measurands trimap3.rawand mnnbrowser.raw(Table A2,) there are correlated CID-MS/MS results from oligomer ion at m/z 1522 depending on experimental variables (Figure A3, Table A3.) Figure A10 correlates MS measurands such as (m/z) values and intensity data on common product ions at m/z 978.06, 1078.87,1178.03,1189.95,1289.95, and 1389.79,respectively.The analysis of (m/z) data shows |r|=1, while relationship between intensity values -|r|=0.9966.The ANOVA data indicates that (m/z) parameters of two sets of variables are not statistically significantly different (Table A4).
Therefore, perturbation of experimental conditions of measurements effects on intensity variables, rather than on m/z ones.

Fragmentation pattern of metacrylic acid
The PMMA standard sample [50] exhibits peaks of product MS ions at mz/ 429, 415, 401, 36,355,341,295,284,281,267,221,207,191,147,112, 97, 83, 71, 57, 43, and 28, respectively.There is good agreement between the data on the current study and standard sample (compare Figures A11 and A12;) thus, achieving |r|=0.9632,correlating between MS measurands [62].MS ions at m/z < 100 allows for testing effects of theoretical level of computations on D QC parameters.There are examined product ions depicted in Figure A13(A).
In addition, it is assessed mechanism of loss of solvent water molecule of MA and intramolecular proton and charge transfer effects (Figure 13

(B).)
Importantly, accurate determination of MS intensity of fragment species enables us to determine positional isomers of monomeric ions, despite their virtually identical MS patterns within low m/z region (m/z 0-100.) For instance, 2-butenoic acid, as can be expected, exhibits similar fragmentation species as MA.There are observed ions at m/z 86, 69, 68, 44, 42, 39, and 29, respectively [63].The obtained results agree with the view that M062X comparing with CCSD(T) method produces reliable and highly accurate theoretical data [64].)

D "
SD data on equation (2) are calculated using experimental variables of product MS ions per short span of scan time.)Experimental MS variables (Tables A5 and A6) of scans 13, 58, 97, 136, and 175 are examined, looking at MS ions at m/z 162, 229, 237, 254, 259, 271, 288, 390, 407, 418, 435, and 453, respectively.The m/z parameters depending on scan are statistically not significantly different (Table A7.)There is mutual linear relation between intensity ratios of product ions (Figure A14; |r|=0.98845-0.99546)per scan; thus, indicating that there are same species per scan time of measurements.

Theoretical data -determination of parameters of Arrhenius's equation
The study focuses the reader's attention on application of stochastic dynamic mass spectrometry to determine molecular structure of polymers; thus, highlighting advantages of this approach to determine reliably 3D molecular conformations, electronic effects, and end-chain chemical substituents of polymers.The research tasks involve determination of D QC parameters of equation ( 3).The calculation steps have been already detailed on preceding papers, dealing with 3D structural analysis of LMWs via complementarily employment in equations ( 1) or ( 2) and (3) [21,22,24].Herein, there is discussed aspects of method associated with vibrational modes of MS species in GSs and TSs (ν i (0) and ν i (s) of equation ( 3).)A key aspect of determining 3D molecular conformations from experimental MS data and D QC parameters is obtaining of ν i (s) results from saddle point.The value of BOMD method, in this context, is that it provides accurately vibrational frequencies at TSs (Tables B1-B3) together with TS thermochemistry of MS species.The approach is proven within the framework of LMWs, as aforementioned.Depending on level of theory, there is obtained exact from perspective of chemical accuracy data on product ions.Owing to the fact, that this study details on polymers, their oligomers, and alkali metal ion adducts, there is tried to maintain balance between accuracy of theoretical data and computational costs; thus, utilising DFT MD methods within the framework of Born-Oppenheimer approach.Therefore, there should be accounted for error contribution to D QC data (Table 1) and affect on method performances assessing relation D " SD =f(D QC ) via chemometrics, due to accuracy of theoretical method, as well.
Despite, as can be seen, statistical analysis of latter function detailed on next subsection determines successfully largest part of product MS ions of molecular models of different types of end-chain substituted polymers and oligomers, thus providing a well-distinguishable and reliable information about degree of statistical significance of linear relations between datasets of MS variables of ions and D QC parameters of species depending on polymer type.

Correlation between theory and experimental mass spectrometric data
The correlative analysis among experimental MS data on product ions involves statistical assessment of experimental measurable variables, in order to account for their statistical similarity.Different m/z parameters belong to species having different electronic structures even within the framework of the same molecular structural units.For instance, there can be tautomeric effects of different protomers of both parent and product ions (consider detail on [18].)As experimental results from Table A6 reveal, there are two mass-to-charge sets of variables at m/z 270.9988±0.02and 271.06836±0.03which are not statistically significantly different (Tables 2 and 3.) (See also nonparameteric two sample Kolmogorov-Smirnov [65], Wilcoxon-Mann-Whitney [66], and Mood's median [67] test data, as well.) Therefore, the peaks belong to the same 3D molecular and electronic structure of ion at m/z 271.Thus, assessment of functional relations D '' SD =f(D QC ) and I Theor SD =f(<I>) of equations ( 2), ( 3) and ( 6) between theoretical and experimental MS data includes D '' SD parameters and average intensity valued (<I>) of datasets of measurands, which are statistically significantly, different.The corresponding D QC parameters belong to different 3D molecular and electronic structural models of parent and product ions according for tautomers, protomers, species results from intermolecular rearrangement or charge transfer effects, respectively.For instance, consider ions at m/z 237 looking at two different molecular structures of Na + -adducts shown as form 237_a and 237_b in Figure 5.
Figure 8 details on chemometric method performances of relationships, above.There is obtained excellentto-exact parameter (|r|=0.99998 ) examining experimental measurands of ions at m/z 183, 229, and 237, as well as theoretical D QC data on form 229_a and 237_b.The employment in D QC value of 3D molecular structure of ion 237_a yields to decrease in |r|-value up to |r|=78935.The result from correlative analysis of relation D ''  SD =f(D QC ) assessing product ions of polymers confirms that model equation ( 2) is an exact function.It agrees well with results reported, so far, from analysis of LMWs [19,20].
Owing to the fact that equation ( 6) is approximated relation derived from equation ( 2) as tool for predicting mass spectra of analytes within the framework of the stochastic dynamics theory the obtained parameter |r|=0.9360 2 provides additional proof of its validity.2), ( 3) and ( 6) of experimental and theoretical mass spectrometric data on ions of PMMA (see Tables 1 and A6); chemometrics

Conclusion
This study has argued for that innovative stochastic dynamic model equations ( 2) and ( 6) (2023) are capable of determining 3D molecular and electronic structures of synthetic polymers; thus, examining high-resolution electrospray ionization mass spectra of poly(methyl-2-methylpropenoat).Owing to the fact that, so far, the model formulas are verified and empirically tested on mainly low molecular weight analytes, it seems to me that one of the main strengths of the current study is the empirical justification and validity of the models to mixtures of synthetic macromolecular oligomers and polymers of the entitled analyte.Furthermore, the correlative analysis between theory and experiment looking at formula (2) yields to excellent-to-exact method performances (|r|=0.99998 .)The latter result makes equation ( 2) directly applicable to exact quantifying of additives to polymers as already highlighted, above, in addition to mass spectrometric based structural analysis of chemically substituted carbon-carbon backbone of polymers and oligomers.Equation (2) appears prominent tool for broad interdisciplinary application to polymeromics, polymer research and materials, as well.

Appendix B
Stochastic dynamic mass spectrometry as an innovative approach to analyse polymers (Quantum chemical data)

Figure 1 .
Figure 1.ESI-MS spectrum of the polymer within m/z  1000-2000; chemical diagrams of possible initial (α) and terminating (ω) ends of chain of PMMA examined in this study

Figures 7 and
A15 depict molecular optimization data via static methods and molecular dynamics.

Figure 7 .
Figure 7. Theoretical (M062X/SDD) molecular optimization of product ion at m/z 237, examining species of type 237 _a and 237 _b : Total energy, E TOT [a.u.] with respect to optimization step number; DFT-BOMD data on same species: Potential energy [a.u.] versus time in trajectory [fs]; optimized 3D molecular structures of product ions

Table 1 .
The D QC data on equation (3) and theoretical intensity parameters I Theor SD of equation (6) of fragmentation ions of polymer

Table 2 .
ANOVA tests results from experimental measurable variables per span of scan time of PMMA at m/z 271 (see TableA6)

Table 3 .
Kolmogorov-Smirnov test data on two sample measurable m/z variables of ion at m/z 271

Table A1 .
Experimental mass spectrometric conditions of measurements of polymers

Table A3 .
Mass spectrometric peak position shown as mass-to-charge (m/z) values and intensity (I) [arb.units]data on the peaks of fragmentation ions of CID-MS/MS spectra of m/z 1522 of msnbrowser.raw and tripam3.raw(TableA2,FigureA3)

Table A4 .
ANOVA test results from MS variables listed in TableA3ad FigureA3

Table A7 .
ANOVA test results from MS variables m/z per scan listed in TableA5

Table A8 .
Normality Shapiro-Wilk test of data on measurands of polymer fragmentation ions per scan of span time (TableA5)

Table B1 .
Thermochemistry (M062X/SDD) of polymer fragmentation ion at GS and TS in gas-phase shown in Figure5; Z-matrixes are given inTable B2

Table B2 .
Z-matrixes of mass spectrometric species of polymer GS and TS states

Table B3 .
Frequency analysis of mass spectrometric species of polymer in ground and transition states