Single-walled carbon nanotubes (SWNTs) are ideally suited for the transport of electrons in one dimension and have commercial potential as nanoscale wires, nanotube field effect transistors (An et al, 2004) and sensors (McEuen, 2003; Tans et al, 1998). Methods for synthesis produce mixtures of SWNTs with different electronic structures, with the electronic structure of any particular SWNT specified by its chirality which is uniquely characterized by a pair of integers (n,m). Covalent chemistries have been developed in recent years that can selectively manipulate the electronic band structures of these SWNTs (Strano et al, 2003), thus increasing the flexibility of these materials in nanoelectronics applications as well as producing additional variables to increase the efficiencies of processes developed to separate SWNTs of different chiralities.

This contribution considers the problem of identifying chirality-specific kinetic rate constants for SWNT reaction networks from experimental data. These kinetic rate constants could be used to design and control chemical reactors to optimize some product quality objective, or, more importantly, the identification of these parameters for several different reacting species could provide insights into the design of molecules to preferentially react with SWNTs depending on their chirality (and hence electronic structure). Such molecular designs could lead to improved selectivity in sensor applications. While many of the analytical results developed in this contribution apply to other chemical reaction networks and potentially even to non-reacting systems, for specificity we will consider the semibatch reactions of SWNTs with diazonium salts, which covalently attach to metallic SWNTs to the near exclusion of semiconducting SWNTs (Strano et al, 2003).

More specifically, this contribution provides a complete kinetic analysis of a reaction network involving SWNTs reacting with 4-hydroxybenzene diazonium, and develops an efficient algorithm to obtain the relative rate constants from absorption data. Further, this particular system motivates the derivation of a mathematical signature for other parameter estimation problems in which the ratios of model parameters can be identified whereas none of the absolute values of the model parameters can be identified. It is argued that reaction networks that consist of large numbers of parallel chemical reactions operating under quasi-steady-state conditions will have this mathematical signature. The rest of this extended abstract provides a more detailed description of the results.

The reaction of SWNTs with 4-hydroxybenzene diazonium salt follows a two-step process (Usrey et al, 2006): a (n,m)-selective adsorption step followed by a covalent reaction step. The kinetic rate constants are different for SWNTs of different chirality (i.e., different values of the integers n and m) due to the differences in electronic structures. This contribution carries out an identifiability analysis for an experimental system considered in a previous paper (Nair et al, 2007), which is a semi-batch reactor in which the diazonium salt is metered in at a precise rate. While this reaction system considers a specific reacting species, it is expected that an accurate determination of the relative values of the kinetic rate constants would provide insights into the structure-reactivity relationship for other types of electron-transfer reactions of SWNTs.

The reaction network consists of a large number of parallel chemical reactions, in which each path in the network is associated with SWNTs of one chirality (i.e., (n,m) pair, and consists of selective adsorption followed by covalent reaction. Through a differential mass balance, we derive an analytical expression for the total number of vacant sites on (n,m)-SWNTs as a function only of (1) the total number of vacant sites on (N,M)-SWNTs where (N,M) is any ordered pair of integers other than (n,m), and (2) the ratio of the adsorption rate constants for the SWNTs of different chirality. The analytical expression holds for any pairs of integers (n,m) and (N,M), that is, the coverages of adsorbed reaction sites on the surfaces of the SWNTs are described by a simple function of the ratio of adsorption rate constants. Insertion of these expressions into molar balances for the SWNTs and some algebraic manipulation results in a set of decoupled ordinary differential equations for the total number of vacant sites for the SWNTs of various chiralities. The decoupled nature of these equations results in very fast integration, so that the rate constants for each (n,m)-SWNT can be estimated very efficiently from the total number of vacant sites for each (n,m)-SWNT which is measured from deconvolution of the absorption spectrum of the SWNT solution (Nair et al, 2006). This analysis requires no assumptions with regard to quasi-steady-state or concerning the relative rates of the adsorption and covalent chemical reactions for the SWNTs.

The nature of setup of the semi-batch reactor is such that the diazonium concentration in the reactor is never allowed to accumulate. Under such a quasi-steady-state condition, the ordinary differential equation for the number of vacant sites on the (n,m)-SWNTs simplifies to an expression that only depends on the ratios of rate constants. Sensitivity analysis indicates that only the ratios of the rate constants can be identified from experimental data collected from such an experimental system. In other words, in a semi-batch reactor containing a mixture of SWNTs of varying chiralities, the value of the rate constant of any (n,m)-SWNT can only be determined relative to the value of the rate constant for any SWNT of different chirality. The experimental system combined with the reactor analysis provides a method of quantifying the relative reactivities of (n,m)-SWNTs, while being unable to identify the absolute value of the rate constants.

The analysis is then extended to characterize the class of parameter estimation problems for which only the ratio of kinetic parameters can be identified. Two closely-related theorems are derived to provide a fingerprint for identifying such systems. For a system dx/dt = f(x), the results show that the absolute magnitude of the kinetic parameters cannot be identified if the vector of kinetic parameters is in the null space of the Jacobian matrix S of f(x). If this null-space condition holds then all of the ratios of kinetic parameters are structurally identifiable if and only if the Jacobian matrix is rank deficient by exactly 1 dimension. That is, this signature characterizes all systems in which the absolute values of the model parameters cannot be determined from data collected from the system whereas the relative values of model parameters can be determined. The jacobian used in the analytical condition can be computed via analytical, automatic (e.g., ADIFOR), or numerical differentiation.

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McEuen, P.L. Phys. World 2000, 13, 31-36

Nair, N.; Kim, W.J.; Usrey, M.L.; Strano, M.S. J. Am. Chem. Soc. 2007, 129, 3946-3954.

Nair, N.; Usrey, M.L.; Kim, W.; Braatz, R.D.; Strano, M.S. Analytical Chemistry 2006, 78, 7689-7696.

Strano, M.S.; Dyke, C.A.; Usrey, M.L.; Barone, P.W.; Allen, M.J.; Shan, H.W.; Kittrell, C.; Hauge, R.H.; Tour, J.M.; Smalley, R.E. Science 2003, 301, 1519-1522.

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