Abstract
Classical theory predicts that branching defects are unavoidable in large dendritic molecules when steric congestion is important. Here we report first experimental evidence of this effect via labelling measurements of an extended homologous series of generations g=1…6 of dendronized polymers. This system exhibits a single type of defect interrogated specifically by the Sanger reagent thus permitting to identify the predicted upturn in the number of branching defects when g approaches gmax and the polymer density approaches close packing. The average number of junctions and defects for each member of the series is recursively obtained from the measured molar concentrations of bound labels and the mass concentrations of the dendritic molecules. The number of defects increases at g=5 and becomes significant at g=6 for dendronized polymers where the gmax was estimated to occur at 6.1 ≤gmax≤ 7.1. The combination of labelling measurements with the novel theoretical analysis affords a method for characterizing high g dendritic systems.
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Introduction
Intense research efforts are devoted to dendritic molecules comprising of repeatedly branched chains known as dendrons. The dendrons can be attached to point-like cores, to linear polymer chains or to surfaces thus giving rise to three leading structural families of dendritic molecules: dendrimers1,2,3,4,5, dendronized polymers (DP)6,7,8,9 and forests10,11,12. Their masses and spans depend on their generation g specifying the number of branching sites along a path between the attachment site and the terminus. The synthesis of ideal dendritic molecules with perfectly regular branching is conceivable for low g values. However, packing constraints rule out such perfect structures at high g when branching defects are unavoidable. The crossover between the two regimes occurs at a theoretically well-defined gmax10,12,13,14,15 discussed below. Since the first reports of well-characterized dendrimers in 1985 (refs 16, 17, 18, 19), thousands of research articles addressed their synthesis, properties and applications1,2,3,4,5,6,7,8,9,20,21,22,23. Recent activity in this area reflects a growing interest in biomedical applications such as bio-imaging, gene and drug delivery24,25,26,27,28. The overwhelming majority of the literature concerns the g<<gmax range and highlights structural perfection. In marked distinction, this work focuses on the vicinity of gmax, the associated onset of structural imperfection and its characterization. It reports first direct evidence for gmax and the associated branching defects. It is enabled by synthesis of a homologous series of DP approaching the vicinity of gmax, their characterization and a theoretical framework for quantifying the number of defects. gmax of a DP occurs at relatively low values thus rendering the synthesis feasible. The near gmax range merits attention for two reasons. First, the existence of the two synthetic regimes is a qualitative signature of dendritic systems that remained essentially unverified and thus of fundamental interest. Second, the exploration of dendritic molecules with g≥gmax is of practical interest for producing densely packed molecular objects with controllable surface properties and an increased range of tunable spans. Note that while the enhanced tuning range is attained at the price of introducing unavoidable defects at g≥gmax, there is little evidence that structural perfection is of practical importance. Apart from the pioneering exploration of the near gmax range, this work presents a theoretical framework for quantifying defect statistics on the basis of labelling reactions. It thus introduces a simple technique for the detailed quantitative characterization of dendritic molecules. This technique is especially useful at high g and high molecular masses (MM) where the traditional analytical methods are challenged. It is particularly suitable for the study of gmax effects diagnosed via defect quantification at high g.
The synthetic regimes outlined above reflect a distinctive form of steric hindrance. The term steric hindrance typically connotes the prevention of a chemical reaction by a bulky chemical substituent within a molecule29. This local effect assumes a novel, global form in dendritic molecules whereby the overall density within the molecule reduces the reactivity of certain groups13. This effect arises in the vicinity of the so-called de Gennes dense packing limit30, at , because the bulkier reaction products cannot be accommodated within the available free volume. In turn, the onset of this effect at gmax occurs because the mass of an ideal dendron grows exponentially with g while its maximum span increases only linearly10,12,13,14,15. For g>gmax the exponential growth of mass, associated with perfectly regular branching, is impossible. Further growth can only be accommodated by imperfect branching structures leading to sterically induced stoichiometry (SIS)30, as was first noted by de Gennes and Hervet in 1983 (ref. 13). Since then, the vicinity of gmax and the SIS were invoked to rationalize deviations of observed MM from their ideal values31,32, as well as synthesis failures33,34,35. It was also discussed as a design guideline for controlling the accessibility of the dendritic interior to guest molecules36,37. With these exceptions, the accumulated evidence of gmax and associated SIS since their prediction in 1983 is scanty and indirect. This state is due to a number of difficulties. On the experimental side, studies confront two problems. First, it is synthetically difficult to approach gmax for the often studied dendrimers where estimates suggest gmax≈10 (refs 13, 38). Second, the observation of gmax effects requires quantification of the frequency of branching defects as a function of g over a wide range of g including the vicinity of gmax. This is experimentally challenging because the performance of standard characterization techniques, such as nuclear magnetic resonance (NMR)39,40,41, mass spectrometry (MS)39 and gel permeation chromatography (GPC)42,43, attain their limits at high g and high MM. While scattering techniques provide information about the dimensions of the dendritic molecules and their density profiles they are not sensitive to branching defects44,45. On the theory side, computer simulations probed the onset of dense packing in perfect dendritic molecules by monitoring properties such as bond strain and equilibration rates38,46,47. However, these studies do not yield direct information on the occurrence and frequency of structural defects because of their focus on ideally branched structures.
In the following, we report a first direct experimental evidence for the existence of two synthetic regimes due to gmax effects and SIS in DP. In particular, we focus on the onset of SIS. Attaining this end required two new inputs: a homologous series of DP with an upper g close to gmax, and a theoretical method of analysing the defect labelling data to quantify the defect statistics.
Results
The two new inputs
The first input was enabled by the divergent synthesis of g=6 DP thus producing an extended homologous series of DP with g=1...6, denoted as PGg (Methods). We investigated two homologous series differing in the polymerization degree of the backbone, N: long DP of N≈1,000 (1,000PGg) and short DP with N≈45 (45PGg). The 1,000PG1...1,000PG5 series was reported already48, while the 1,000PG6 and the 45PG1...45PG6 series are new (Methods). As we shall discuss, the differences between the two have a key role in identifying the gmax effects and the onset of SIS. The second, theoretical input is the result of an analysis yielding a recursive equation specifying the average number of junctions and termini from measured quantities.
With these two inputs at hand, the exploration of gmax effects in this system is facilitated by three features (a) gmax in long DP occurs at lower g range as compared with dendrimers10,49. Current estimates for the particular DP studied suggest 6.1gmax7.1 (ref. 50) and thus at the boundary of the g=1...6 homologous series. (b) The DP as synthesized exhibits a single type of defect with no side reactions. In particular, the defect is associated with the occurrence of non-reacted primary amine groups. (c) The number of defects can be quantified via the ultraviolet (UV) absorbance or fluorescence of a label specifically binding to these defects. In this study, the label used is the Sanger reagent51,52. In turn, these last two features enable the theoretical analysis of the labelling data of the homologous series of DP in order to quantify the number of defects for different g. As we shall discuss, the number of defects increases significantly for 1,000PG6 thus suggesting the onset of SIS in the vicinity of gmax. Importantly, the effects are evident only in the long 1,000PGg but not in the short 45PGg. This supports the SIS interpretation as defect accumulation due to imperfect reaction conversion should affect all DP irrespective of N. In contrast, SIS effects in the g range explored are envisioned only in long, locally cylindrical DP because of their lower gmax. They are not expected in the 45PGg series because of their higher gmax arising since short DP adopts a dendrimer-like, near spherical, shape traceable to backbone end effects.
Synthesis
The divergent synthesis of a dendron proceeds from a f-functional root attached to a point-like core, to a linear polymer chain or to a surface (Fig. 2 and Supplementary Fig. S1). The roots are reacted with X-functional dendronization units (D units), having one reactive functionality and X−1 blocked, non-reactive, functional groups. In the next step, the blocked groups of the D units bound to the root are activated and their deblocked groups reacted with newly added D units. A g generation dendron is generated by g iterations of the deblocking reaction sequence. The generation g of the dendron thus specifies the maximal number of branching sites, junctions, along a strand joining the root to a terminal group. In the following, we focus on D units having the overall structure of a dendron junction or, equivalently, a g=1 dendron. In other words X is the functionality of the D unit, as well as the number of ‘arms’ emanating from a junction. In an ideal, structurally perfect g generation dendron each terminal junction bears X−1 blocked functionalities and every interior junction involves X bonds between chemically linked D units. Stated differently, the ideal structure contains no free functionalities that is, all functionalities are either blocked or covalently bound to another D unit. In a non-ideal dendron, some of the unblocked functionalities are not reacted and remain free and active (Figs 1 and 2). Additionally, some may undergo side reactions rendering them inactive.
An iterative relationship between the numbers of junctions and of the free or blocked ends
As the theoretical analysis is applicable to dendrimers as well as DP of different chemistries, we formulate it in general terms. Our theory considerations concern ‘elementary dendritic motifs’: a dendrimer, a repeat unit of a DP comprising of a root with the attached dendrons and the corresponding unit in a dendronized surface. We focus on the simplest case where there are no side reactions and the defects consist of free non-reacted functional groups, ‘free ends’. In this case, the structure of generation g dendritic motif is specified by the number of junctions ng, together with the number of blocked and free ends denoted respectively by and . The total number of ends, free or blocked, in generation g dendritic motif is =+. In turn, =f+(X−2)ng as the functionality of the root is f and each additional junction eliminates one functionality while contributing X−1 additional ones. The number of additional junctions generated with the gth synthesis step is denoted by Δng≡ng−ng−1. In a generation g dendritic motif there are ng−1 ‘non terminal’, inner junctions. All Δng ‘new’ junctions are terminal junctions comprising of attached D units each with (X−1) blocked functionalities. Accordingly, the number of blocked terminal functionalities in a dendritic motif of generation g is =(X−1)Δng. Combining =(X−1)(ng−ng−1) with the earlier relationships yields the key recursive equation
relating ng, ng−1 and in a dendritic motif of a given X and f (Fig. 3). It applies to the values characterizing individual dendritic motifs, as well as to ensemble averages denoted by ‹..›. Our key result is based on the utilization of Equation (1) to relate ‹ng›, ‹ng−1› and ‹› of a homologous series of . In this case, it is possible to iteratively solve equation (1) and to obtain from labelling data providing partial information on ‹›, ..., , together with the initial values ‹n0›=‹›=0. Importantly, the complete set of ‹ng› thus obtained fully specifies the corresponding ‹Δng›, ‹› and ‹›.
Theoretical analysis of a labelling experiment
To obtain ‹ng› for the homologous series, we utilize labelling experiments where we measure two concentrations. The first is the molar concentration of labels bound to dendritic molecules of each generation, . For the case of efficient labelling reagent considered, we assume that all free ends are labelled and the number of labelled ends per dendritic motif is =. We note that this assumption is tenable only for ggmax. For g>gmax the free ends at the dendritic interior may be inaccessible to the labels thus leading to <. The onset of this effect depends on the size distribution of dendron voids, the size of the label and its interactions. The second measured quantity is the mass concentration of dendritic molecules w/V as specified by the weight of the dry labelled sample, w, and the solvent volume, V. In the following we convert the mass concentration to the molar concentration where MY is a typical MM of a junction. MY accounts for the contributions of both terminal junctions, bearing the two blocking groups, and internal junctions with no blocked groups (Fig. 2): MY=+(X−2) where is the MM of a chemically bound D unit within the ‘interior’ of the dendron and is the increment of MM of a blocking group bound to a deblocked D unit (Fig. 1). and are both proportional to cg, the molar concentration of labelled dendritic motifs of generation g: =cg‹› and where ‹› is the initially unknown average MM of a labelled dendritic motif of generation g. Their ratio,
is thus independent of cg. Ug is a rough estimate of ‹›/‹ng› and it relates ‹› to the yet to be specified ‹›. In turn, ‹›/MY is a linear function of ‹ng› and ‹ng−1› with known coefficients Alabelled, Blabelled and Clabelled (Methods and Fig. 1). This, together with equations (1) and (2) yields an iterative equation for ‹ng›,
with the boundary condition ‹n0›=0, as specified by U1,..., , the three coefficients determined by the chemical structure, as well as f and X (Methods, Fig. 1). For the particular DP investigated in this study, f=1 and X=3, hence Alabelled≈0.8117, Blabelled≈0.3766, and Clabelled≈0.6798 (Methods). Furthermore PG1 was obtained by polymerization of appropriate P units with f=1-functionality bound to a blocked D unit (Fig. 2). Consequently, PG1 can be considered structurally perfect, corresponding to U1=0. Under these circumstances, the first two solutions of equation (3) are accordingly ‹n1›=1 and ‹n2›=[3−U2(Blabelled+Clabelled)]/(1+U2Alabelled). Higher g terms are obtained iteratively along the same lines.
Analysis of the labelling measurements
The results of the above analysis of the DP homologous series labelling data are depicted in plots of ‹ng›, ‹› and ‹› versus g for both the 1,000PGg and 45PGg series (Figs 4 and 5). The 45PG1...45PG6 and 1,000PG1...1,000PG5 are found to exhibit the behaviour of an ideal DP,
equivalent to equation (3) when Ug=0 for all g. In particular, ‹›/2 is indistinguishable from the number of terminal junctions in an ideal structure, =f(X−1)g, and there are essentially no free ends, . In contrast, the 1,000PG6 results deviate from this trend: ‹n6›<, ‹›< and (Table 1). The onset of the deviations actually occurs at 1,000PG5 as can be seen from the ‹›/‹› versus g plot (Fig. 6) that amplifies small deviations that are hard to discern in the ‹› versus g plot. The upturn in the number of defects occurs below the estimated gmax of this system, 6.1<gmax<7.1 (ref. 50). However, the onset of packing effects is expected below gmax because the densification of the system slows down transport with corresponding effect on the labelling kinetics. Importantly, the observed deviations occur only for the locally cylindrical 1,000PGg series. There are no comparable effects for the 45PGg homologous series, which exhibit ideal DP behaviour as specified by equation (4). This is an important observation supporting the interpretation of the data in terms of gmax-related SIS effects. Two arguments are involved. The first concerns the N dependence of gmax and the onset of SIS effects. For long locally cylindrical DP, the current estimate of gmax of this chemistry vary in the 6.1<gmax<7.1 range50 and are thus consistent with onset of SIS at In contrast, short DP at high g exhibit dendrimer-like configurations because of backbone end effects. This suggests higher gmax closer to 12.7<gmax<14.1 estimated for dendrimers of similar chemistry50. Accordingly, gmax effects are not expected for 45PG1...45PG6. The second argument concerns the merits of an alternative explanation of the defect statistics in terms of imperfect conversion of the dendronization reactions. When excluding gmax-dependent SIS effects, this second mechanism is expected to affect both 45PGg and 1,000PGg homologous series. The absence of such trend supports the interpretation in terms of SIS effects.
Discussion
Utilizing labelling technique we obtained evidence for the onset of SIS gmax effects in long DP of g=6gmax close to 6.1<gmax<7.1. This result is the first systematic confirmation of packing effects on the synthesis of dendritic molecules since their prediction in 1983. The attainment of g≈gmax is also a first step towards the methodical exploration of g>gmax dendritic molecules of interest as dense molecular objects with tunable size. Finally, the labelling technique, together with the theoretical framework reported, is a promising characterization technique for a significant family of dendritic molecules. As we shall discuss, it balances advantages, due to light instrumental and computational requirements, with shortcomings regarding its range of applicability. The labelling technique is applicable when four conditions are satisfied: (i) A homologous series of dendritic molecules is available. This is typically the case when utilizing divergent synthesis. (ii) The dominant defects are non-reacted, deblocked terminal groups. This is often the case for divergent synthesis using g=1 dendrons as D units. This condition is realized, for example, in Denkewalters dendritic peptides17, Newkomes arborols18 and in Simaneks triazin-based dendrimers53. (iii) The defects can be efficiently interrogated by a labelling reagent. There is no universal label and the choice of the reagent should be customized to the synthetic chemistry used. When NH2 groups are involved one may use the Sanger reagent51,52 or Dansyl54,55,56. The Sanger reagent is less prone to aggregation and its smaller size is advantageous at high g. (iv) The labelling method can quantify all free ends for 1≤ggmax, that is, ‹›=‹›. For g>gmax, it provides an efficient probe of exterior, accessible, free ends. However, the labels may encounter steric hindrance when binding defects situated within the dense dendritic interior that is,, ‹›<‹›. When these four requirements are met, the labelling method yields ‹ng›, ‹Δng›, ‹› and ‹› for thus providing a complete characterization of the average structure. In particular, it specifies the average MM of the dendritic motif via equation (5) or its counterpart for the non-labelled moiety. MS when applicable, is free of these limitations31,39. However, it requires a dedicated facility and significant computational effort to deconvolute the spectra. In marked contrast, the labelling approach based on the Sanger method requires a ultraviolet spectrophotometer and minimal computational effort. With regard to MS and NMR, one should also note difficulties in characterizing high g dendritic molecules in general and DP in particular. These difficulties are even more pronounced for GPC.
The interest in dendritic molecules is motivated in part by the vision of single-molecule colloidal particles of well-defined chemical structure. This direction is illustrated by 17,600PG5, a single-molecule comparable in length and width to the potyvirus family and common cytoskeleton fibrils48,57. If one adheres to this point of view, it is important to establish gmax because it signals the onset of SIS thus setting the upper boundary of synthetically attainable ideal dendritic molecules. However, the ‘post gmax’ regime may afford opportunities in spite of the unavoidable structural defects. Along the lines of the de Gennes–Hervet argument one may hypothesize that the structure of g>gmax dendritic molecules is controlled by packing constraints resulting in dense molecules with well-defined dimensions. The investigation of this hypothesis again requires knowledge of gmax, as well as efficient defect quantification methods for the gmax range. Note that defects affect properties of dendritic molecules that depend on the number of end-groups, such as solubility. These observations confront a current lack of systematic observations of gmax effects traceable to synthetic and characterization difficulties facing the exploration of the gmax range. In this context, our work pioneers the synthesis of near gmax DP and the quantitative exploration of the branching defects associated with SIS. It thus initiates the systematic investigation of the g≳gmax regime.
Methods
Synthesis of deblocked de-1,000PG5
Some of the starting material, denoted by 1,000PG5, may have undergone charge-assisted, shear-induced main chain cleavage58, thus resulting in product of lower N. To a slowly- stirred, freeze-dried powder of 1,000PG5 (0.20 g, 0.018 mmol repeat units) in a round bottom flask was dropwise added trifluoroacetic acid (TFA) (25 ml) and methanol (1 ml) at 0 °C. The reaction mixture soon turned homogeneous and was stirred at room temperature for 1 h. Then methanol (20 ml) was added and the mixture was evaporated in vacuo. This methanol addition and evaporation procedure was repeated twice, thereafter the residue was dissolved in water (15 ml) and carefully lyophilized to yield de-1,000PG5 as a powdery white solid (0.21 g, 100%). 1H NMR (500 MHz, DMF-d7): δ=0.90 (br, 18H, CH3), 1.33 (br, 45H, CH3), 2.01–2.19 (br, 326H, CH2), 3.20–3.49 (br, 252H, CH2NH), 4.12 (br, 252H, CH2O), 6.42 (br, 75H, Ph), 7.08 (br, 142H, Ph), 8.41 (br, 30H, Ph, NH). Note: It is essential to avoid exposure of the de-1,000PG5 to shear forces because the chains can suffer a charge-assisted, shear-induced main chain cleavage58.
Synthesis of blocked 1,000PG6
To a dimethylformamide (DMF) solution (250 ml) of de-1,000PG5 (200 mg, 0.0174, mmol repeat unit) at −5 °C was added triethylamine (88 mg, 0.87 mmol) and N,N-dimethylaminopyridine (DMAP) (30 mg, 0.25 mmol). A solution of the D unit, the active ester dendron DG1 (3.15 g, 5.57 mmol) was added in six portions over 20 days. During the addition of each portion, the reaction mixture was cooled to −5 °C, then slowly warmed to room temperature and stirred for 3–4 days. After the addition of all the active ester dendron DG1, the reaction mixture was left stirring for another 10 days, then concentrated in vacuo. The residue was dissolved in 20 ml of dichloromethane (DCM) followed by column chromatography purification (eluent: DCM, Rf=0.1). This produced a beige gel, which was freeze-dried from 1,4-dioxane (40 ml) to yield 1,000PG6 as a powdery, white solid (78 mg, 20%). 1H NMR (500 MHz, DMF-d7): δ=0.91 (br, 45H, CH3),1.36 (br, 576H, CH3), 1.89–2.02 (br, 273H, CH2), 3.04 (br, 134H, CH2), 3.18 (br, 153H, CH2), 3.62 (br, 99H, CH2), 4.08 (br, 252H, CH2), 6.19 (br, 66H, CH), 6.58 (br, 79H, CH), 7.05 (br, 144H, CH), 8.16 (br, 45H, NH). Calcd for (C1143H1654N126O318)N: C, 61.71; H, 7.49; N, 7.93. Found: C, 61.19; H, 7.20; N, 7.52.
Synthesis of 45PG1…45PG6
45PG1 was synthesized by a reversible addition-fragmentation chain transfer (RAFT) protocol according to the reported procedure59. Divergent synthesis of 45PG2...45PG6: to a solution of deprotected polymer precursor de-45PG(g-1) in DMF was added 4-dimethylaminopyridine (DMAP, cat.) and triethylamine (TEA, 2 eq. per amine) at −5 °C. Dendron active ester DG1 (5 eq. per amine) was added in three portions over a total predetermined time (4–15 days). Each portion was added at −5 °C, followed by a stirring at room temperature for 1–5 days. After the addition of the last portion and stirring, the reaction mixture was concentrated in vacuo. The residue was dissolved in DCM and purified by column chromatography (silica gel, DCM eluent) to give a beige gel, which was lyophilized from 1,4-dioxane to yield the product 45PGg as a white powder (71–88%). Normalized GPC elution curves of 45PG1...45PG6 are provided in Supplementary Fig. S2.
Labelling the DP
To a well-stirred solution of blocked DP (5.0 mg) in 1,1,2,2-tetrachloroethane (0.3 ml) was added 0.1 M NaHCO3 solution (0.043 ml), and a solution of Sanger reagent (0.4 mg, 0.3 eq. per Boc) was added in 1,1,2,2-tetrachloroethane (0.04 ml). The reaction mixture was stirred at 65 °C for 3 h, and then cooled to room temperature. 1,1,2,2-tetrachloroethane (2 ml), water (2 ml) and citric acid (1 mg) were added to the mixture. The organic layer was separated, washed by water (1 ml) and brine (1 ml), and concentrated in vacuo. The residue was dissolved in tetrahydrofuran (1 ml) and precipitated into methanol/water (4:1). This procedure was repeated four times to yield the labelled polymer in the form of a yellow solid (3.5 mg, 70%).
Ultraviolet absorbance measurements
Measurements (Fig. 7) were performed on a Perkin Elmer Lambda 20 and a Jasco V-670 ultraviolet–vis spectrometer using 1 mm or 2 mm quartz cells, respectively. The labelled 1,000PG6 was dissolved homogeneously in 1,1,2,2-tetrachloroethylene. The extinction coefficient ε=1.64 × 104 l mol−1 cm−1 (ref. 52) was assumed for 2,4-dinitroaniline moieties. Their concentration was calculated using the Lambert–Beer law (clabelled=A/εl), in which l=0.1 cm (1,000PGg) or l=0.2 cm (45PGg) denotes the interior width of the quartz cell and A the absorbance at wave length λ=384 nm (Fig. 7).
Derivation of equation (3)
The dimensionless coefficients Alabelled, Blabelled and Clabelled occurring in
are determined uniquely by the chemical structure
All masses entering equation 6 had been introduced in Fig. 1. Equation (5) with equation (6) follows from the mass accounting relationship =M0+ng++ upon substituting =(X−1)(ng−ng−1) and allowing for =+=f+(X−2)ng. Insertion of ‹›=Ug[Alabelled‹ng›+Blabelled‹ng−1›+Clabelled] into equation (1) leads to equation (3).
Error estimates
The masses of the labelled polymer samples, w, and masses of the solution, ws, were measured up to a precision of δw=0.1 mg and δws=10 mg, respectively, taking into account both instrumental and evaporation errors. The volumes of the polymer solutions are calculated from the measured masses, V=ws/ρ with ρ=(1.597±0.001)g cm−3 assuming the density of the dilute polymer solution is identical with that of the pure solvent (Table 2). The molar concentrations of labels =A/ε l had been determined from the ultraviolet adsorbances A that come with a maximal relative error of 2% for the 45PGg series (measured with Jasco V-670 (ref. 60)), and 4% for the 1,000PGg series (Perkin Elmer Lambda 20 (refs 48, 15)). The error of the denominator follows from δε≈100 (ref. 52) and δl≈1 micron. The relevant ratios
furthermore involve our reference mass of a typical branching unit, MY=(350.415±0.001) g mol−1. The relative error δUg/Ug thus follows immediately from the reported values in Table 2 and their above-mentioned errors by summing up the relative errors for all quantities on the rhs of equation 7.
The calculation of ‹ng› via the iterative relationship (equation 3 with f=1 and X=3)
involves the dimensionless constants Alabelled, Blabelled and Clabelled. As the chemical structure of the ideal dendron is known exactly, the relative error of these three coefficients is negligible. ‹n1›=1 being error free as discussed within the Results section. The error of ‹ng› remains to be determined for given value and error of the preceding ‹ng−1› and the calculated Ug±δUg. Fortunately, the errors δUg occur in combinations of the form 1+AlabelledUg. Importantly, because δUg<<1 and because all coefficients Alabelled, Blabelled and Clabelled are of order unity, the relative error δ‹ng› is mainly determined by (X−1)δ‹ng−1› alone, and does not grow exponentially. Furthermore, for g=4 or higher, when ‹ng−1›>>1, the last bracket in the nominator of equation 8 becomes clearly irrelevant.
If we denote the absolute error of ‹ng› with δ‹ng› its relative error is
with coefficients
For example (g=2):
As ‹n1›=1, and this yields ≈1.2 × δU1 for the numerical values of Alabelled, Blabelled and Clabelled that are specified after equation 3.
Additional information
How to cite this article: Zhang, B. et al. Synthetic regimes due to packing constraints in dendritic molecules confirmed by labelling experiments. Nat. Commun. 4:1993 doi: 10.1038/ncomms2993 (2013).
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Acknowledgements
We thank Dr Bernd Bruchmann, BASF and Prof. Dr Renato Zenobi, ETH Zürich for helpful discussions.
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A.D.S., A.H. and M.K. designed research. B.Z. and H.Y. performed experiments. M.K. developed model. M.K. and B.Z. analysed data. A.H. wrote the paper.
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Zhang, B., Yu, H., Schlüter, A. et al. Synthetic regimes due to packing constraints in dendritic molecules confirmed by labelling experiments. Nat Commun 4, 1993 (2013). https://doi.org/10.1038/ncomms2993
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DOI: https://doi.org/10.1038/ncomms2993
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