Uwe Manthe
email: uwe.manthe@uni-bielefeld.de

Accurate calculations of reaction rates: Predictive theory based on a rigorous quantum transition state concept

Keywords: Rate constants, Reactions, Quantum dynamics, Transition state, Tunneling

Accurate calculations of reaction rates: Predictive theory based on a rigorous quantum transition state concept

Introduction

The last two decades have seen significant progress in the accurate simulation and detailed understanding of elementary chemical reaction processes. Considering the most fundamental and precise level of theory, the accurate description even of seemingly simple chemical reactions is a challenging task: quantum effects as tunneling or zero point energy have to be incorporated in the description of the motion of the atoms during the reaction process, particularly if reactions including the transfer of hydrogen atoms or protons are considered. Then the delocalized nature of quantum mechanics, most prominently reflected in the Heisenberg uncertainty principle, requires the knowledge of extended regions of the underlying electronic potential energy surface (PES) of the molecular system. Furthermore the complex nature of the chemical reaction process typically results in strong couplings between several degrees of freedom and requires multi-dimensional PESs and quantum dynamics simulations. Consequently, the use of dimensionality reducing approximations is highly problematic and full-dimensional quantum mechanical calculations are required if one aims at reliable theoretical predictions.

The first rigorous, i.e. full-dimensional quantum mechanical, calculations studying a chemical reaction was published in 1976 [1]. Here the simple-most chemical reaction, the H + H 2 → H 2 + H hydrogen exchange, was investigated. Further progress from triatomic reactions towards polyatomic reactions faced a central problem: the exponential scaling of the numerical effort of grid or basis set based numerical techniques with the dimensionality. The first full six-dimensional quantum calculation for four atom reactions only appeared in the early 1990's [2][3][4] studying the H 2 + OH → H + H 2 O reaction. The step beyond tetratomic reaction was then taken already in 2000 [5] when full-dimensional quantum dynamics calculations for a six atom reaction, H + CH 4 → H 2 + CH 3 , were presented for the first time. It should be noted that the first such calculations for a tetratomic reaction [2] as well as for a six atom reaction [5] explicitly focused on the calculation of reaction rates. Thus, rigorous reaction rate calculations have a pioneering role in the quest towards the accurate description of increasing complex reactions. Besides the focus on selected observables as reaction rates [2,5] or initial state-selected reaction probabilities [3,4,6,7] (the first full-dimensional quantum dynamics calculation of initial state-selected reaction probabilities for a six atom reaction appeared just recently [6,7]), the development of efficient schemes for the multi-dimensional wave packet propagation, particularly the multi-configurational time-dependent Hartree (MCTDH) approach [8,9], contributed significantly to the progress.

Thermal rate constants are generally the magnitudes most relevant for a chemist investigating a reaction. A theoretical approach focusing on the accurate calculation of rate constants is thus well adapted towards the needs of research in chemistry. Considering chemistry in general, transition state theory (TST) provides the central framework dominating a chemists thinking about reaction rates. It is therefore particularly interesting to note that the above mentioned rigorous reaction rate calculations [2,5] employ concepts [10][11][12] which can be viewed as a rigorously correct, quantum mechanical generalization of the transition state (TS) approach. Thus, these reaction rate calculations do not only provide accurate reaction rate data but can also help to understand quantum effects in reaction dynamics using intuitive pictures derived from a quantum transition state concept.

The present article reviews research focusing on the accurate calculations of reaction rates employing rigorous theory. It does not attempt to cover approximate approaches based on (quantum) transition state ideas. There is a variety of such approaches and detailed comparisons of their results with rigorous reaction rate calculations reviewed here have been published [13][14][15][16]. Furthermore, we will restrict our attention to reaction proceeding via potential barrier or through a similar dynamical bottleneck. In these situation the passage of the reaction barrier is the central step of the reaction process and a transition state picture provides physically adequate description. Elementary reactions proceeding not via barriers but via potential wells causing long-living intermediates show very different dynamical behavior. Here transition state based concepts are physically inadequate. The efficient treatment of such reaction requires different concepts, e.g. combined statistical and capture approaches [17], which will not be reviewed here.

Accurate reaction rate calculations can be divided in three tasks which have to be addressed: first electronic structure calculations must accurately compute electronic energies at given geometries, then a multi-dimensional PES has to be constructed based on the electronic structure data, and finally thermal rate constants must be obtained from accurate quantum dynamics simulations on this PES. The methodology of ab initio electronic structure calculations is a highly developed area of research with distinct character, therefore the present review will not address this part required in reaction rate calculations explicitly. It only should be noted that highly accurate electronic structure calculations are required to predict thermal rate constants of chemical reactions: due to the e -E/kBT Boltzmann factor present in the theory of thermal rates, energy differences are always measured relative to the thermal energy k B T . Relevant predictions thus require to computed electronic Theoretical concepts, results, and important implications of rigorous reaction rate calculations are reviewed in Sect.2. First, the central steps in the development of the quantum transition state concept are described. Then the advance in applications from triatomic systems to six atom reactions as well as reactions on surfaces and in condensed phase is reviewed. Relevant techniques which contributed to this progress will be briefly explained subsequently. Finally central conclusions relevant for a wider chemical community are discussed: When can quantum effects expected to be important? What is the impact of zero point energy and tunneling? Is the separation of internal motion and overall rotation a good approximation?

The construction of multi-dimensional PESs required in accurate reaction rate calculations is another central topic which is reviewed in Sect.3. The methods and ideas which formed the basis for the tremendous progress in this area in the last decade are first described in a general context. Then specific aspects of the construction of PESs for reaction rate calculations are discussed. It is highlighted that efficient schemes for the PES constructions in the reaction rate context are "semidirect" in character, relate to the quantum transition state concept, and require only a moderate number of ab initio data points even for polyatomic reactions showing many degrees of freedom.

The article closes with a final section discussing conclusions and future perspectives of reaction rate calculations and the investigation of mode-specific reaction dynamics based on the quantum transition state concept.

Reaction rates: quantum dynamics

Rigorous quantum transition state concept

Quantum scattering calculations computing the scattering matrix yield a fully state-specific description of chemical reaction processes. Wave packet dynamics calculations which simulate the dynamics of a wave packet moving from the reactant asymptotic region towards the product asymptotic region can provide a similar wealth of information about the reaction process. Aiming at the calculation and understanding of the thermal rate constant k(T ) of a chemical reaction, a different strategy is more rewarding: k(T ) is determined by the passage of the reaction barrier and a state-specific description referencing explicitly to quantum states of the reactant or product is not required to compute or physically interprete k(T ). Concepts taken from statistical mechanics and dynamical simulations restricted to the barrier region provide a more intuitive and numerically more efficient description. Classical transition state theory (TST) is based on these ideas and the importance of TST for a general chemists understanding of elementary reaction processes can hardly be overestimated. However, TST is an approximate theory and the present review focuses on the rigorous calculation of reaction rates.

Flux correlation functions, introduced in their quantum mechanical formulations by Yamamoto [20] and Miller and coworkers [10,11], provide the rigorous theoretical framework to compute reaction rates based on statistical mechanical concepts without reference to asymptotic reactant or product states. There is a variety of specific flux correlation functions using energy as well as time-dependent representations. The connection to TST can nicely be seen employing a flux position correlation function [10]:

k(T ) = 1 Q r (T ) lim t→∞ tr e -Ĥ k B T F e i Ĥt h e -i Ĥt (1)
(Please note that = 1 is used throughout this article.) Here Ĥ is the Hamiltonian, Q r (T ) is the partition function of the reactants, and h specifies the dividing surface which separates reactant and product geometries. h equals unity on the product side of the dividing surface, and vanishes on the reactant side. The flux operator F = i[ Ĥ, h] measures the flux through the dividing surface. Eq.( 1) implies a transition state based interpretation of the reaction process: considering the full thermal ensemble, 1 Qr(T ) tr e -Ĥ k B T ... , the thermal rate constant is given by the flux through the dividing surface, which will finally end up as product in the infinite future (for a more detailed explanation of the theory and its connections to TST see, e.g., previous reviews [21][22][23][24][25]). The numerically most efficient approaches typically do not compute the thermal rates directly but employ the cumulative reaction probability N (E) which is related to k(T ) via Boltzmann averaging:

k(T ) = 1 2π Q r (T ) dE N (E) e -E k B T . (2)
Flux correlation functions for the calculation of N (E) can be derived based on the flux correlation formula of Miller et al. [11],

N (E) = 2π 2 tr F δ( Ĥ -E) F δ( Ĥ -E) . (3)
As seen in Eqs.( 1) and ( 3), the calculation of k(T ) or N (E) via flux correlation functions requires to compute averages over thermal or micro-canonical ensembles. These averages are represented by the traces present in the above expressions. Their evaluation presented a major computational problem which initial work utilizing flux correlation functions faced. The straightforward evaluation of the trace employing a complete basis set ψ j , tr( Â) = N j=1 < ψ j | Â|ψ j >, is infeasible even for medium sized problems due to the large size N of the basis set required. Using the specific properties of the flux operator [26] to avoid tracing over the reaction coordinate can not really solve this problem: considering a f-dimensional problem tracing with respect in the other (f-1) coordinates still has to be done. Consequently, early quantum calculations using flux correlation functions [26][27][28][29][30][31][32][33][34][35][36][37][38] were restricted to systems not exceeding three atoms.

While flux correlation functions immediately relate the reaction rate to the quantum dynamics in the vicinity of the reaction barrier and thus provide a first connection to transition state idea, another concept taken from transition state theory can be used to address the problem posted by the thermal or micro-canonical ensemble averages required. This has first been demonstrated in Ref. [12]: rewriting Eq.(3) as N (E) = tr( P (E)) a hermitian, positive semidefinite operator P (E) called reaction probability operator had been introduced. Investigating the H +H 2 reaction for vanishing total angular momentum J = 0, P (E) was found to have only a very small number of non-vanishing eigenvalues which can straightforwardly be interpreted as the contributions of the different vibrational states of the activated complex. Since the number of relevant vibrational states of the activated complex contributing to the reaction rate at relevant temperatures or energies is typically small, only few quantum states contribute relevantly to the thermal or micro-canonical ensemble averages in the flux correlation functions (for J = 0). Thus, the operator within the trace is of low rank and its eigenvalues can efficiently be computed by iterative diagonalization employing a Lanczos scheme. Building on this conceptional basis and sparked by the successful application of these ideas to the four atom reaction 39], a variety of approaches for the rigorous and efficient calculation of reaction rates had been developed subsequently [40][41][42][43][44][45][46][47][48][49][50][51][52]. Important steps taken in the development towards the presently employed approaches should be noted: First, the theory has been reformulated in time domain and wave packet propagation techniques have been employed [40,41]. Second, the schemes for the construction of the wave packets describing the vibrational states of the activated complex could be separated from the real time propagation of the wave packets required to compute N (E) and k(T ) [43][44][45]. The developments established two similar schemes as the present state of the art: the thermal flux eigenstate based approach of Refs. [48,49] (which uses either MCTDH wave packet propagation [48,53] or standard wave packet propagation schemes [49]) and the transition state wave packet approach of Refs. [43] (used with standard wave packet propagation schemes).

H 2 + OH → H + H 2 O [2,
Finally one aspect which has been ignored up to now must be added. The above discussion focused on internal motion and mainly considered calculations for vanishing total angular momentum (J = 0). Then the number of relevant vibrational states of the activated complex tends to be small for relevant temperatures since the vibrational frequencies are usually much larger than the thermal energy. This situation changes dramatically if rotational motion is considered. Rotational spacings are typically much smaller than thermal energies. Thus, if rotation is included explicitly in the calculation, a large number of rovibrational states has to be considered. Propagating a wave packet for each of these states can post a significant challenge. However, this problem was overcome by utilizing a rigorously correct statistical approach [54,55] which can correctly account for rotationally excited states without strongly increasing the number of wave packets to be propagated compared to the J = 0 case.

From triatomics to polyatomic reactions

Prior to the step in method development taken in the early 1990s [12], early reaction rate calculations studied selected triatomic reactions. The full-dimensional calculations for the [2,39] then presented the first benchmark application utilizing the rigorous quantum transition state concept to its full extend. Here the cumulative reaction probability N (E) has been obtained in converged full-dimensional quantum mechanical calculations. The calculations were restricted to total angular momentum J = 0 and the thermal rate constant was obtained in the J-shifting approximation [56] which implies separability of internal motion and overall rotation in the barrier region. Comparison with experiment showed deficiencies in the Schatz-Elgersma PES [57,58] employed.

H 2 + OH → H + H 2 O reaction
After this initial success, the following work extended and further developed the methodology. In these studies, first different triatomic reactions as H 2 + Cl → H +HCl [59][60][61] and O( 3 P )+HCl → OH +Cl [45,47,54,55,62] were investigated. Here also contributions for total angular momentum J > 0 were considered and the coupling of internal motion and overall rotation were studied. Later reaction rate calculations for the four atom reactions [49,66], and HCl + OH → Cl + H 2 O [67] followed. Some of these studies [48,54,55,60,61,65,67] employed the MCTDH approach [8,9] in these calculations.

H 2 + OH → H + H 2 O [48, 63-65], H 2 + CN → H + HCN
Based to the combination of MCTDH wave packet propagation and reaction rate calculations via flux correlation functions, larger reactive systems could be investigated in the early 2000s: the first converged full-dimensional quantum dynamics calculation for a six atom reaction studied N (E) and k(T ) for H+CH 4 → H 2 +CH 3 [5,[68][69][70]. Slightly later the reaction O + CH 4 → OH + CH 3 [71] was investigated on the same level of theory. It should be noted that previous studies of triatomic and tetratomic reaction yielded a detailed understanding of the effects of the overall rotation on thermal rate constants, so that these calculations could safely be restricted to total vanishing angular momentum and employed the J-shifting scheme [56].

While the above calculations could treat the quantum dynamics of the reaction process rigorously, still the underlying PES used in these calculations [72,73] were not accurate and resulted in significant differences between experiment and theory. Therefore the further development considered the construction of accurate full-dimensional PES which will be discussed in some more detail in Sect.3. A new accurate PES was constructed for the H + CH 4 → H 2 + CH 3 reaction and truly accurate reaction rates were obtained performing rigorous full-dimensional quantum dynamics calculation on this PES [18,19]. Considering careful error estimates for all steps of the theoretical description, one could conclude that the accuracy of the computed rates was at least comparable to the accuracy of the experimental data available (see Fig. 1 for an Arrhenius plot of the computed and experimental rate constants). Later on, also k(T )'s and N (E)'s for isotopic analog reaction D + CH 4 → HD + CH 3 [74] and the reverse reactions 75] were calculated on the same level of theory.

H 2 + CH 3 → H + CH 4 and D 2 + CH 3 → D + CH 3 D [
Beside the investigation of gas phase systems (here further calculations ranging from triatomic [76,77] to hexatomic reactions [78][79][80] could be mentioned) also rates for process on surfaces [81,82] and reactions in condensed phase [83,84] were studied based on the quantum transition state concept. The dissociative adsorption of N 2 on a stepped Ru-surface was studied by 6D quantum dynamics calculations for a rigid Ru-surface [82] and the effect of surface motion on in the system was considered on TST level [85]. The rates for hydrogen diffusion on a Cu surface where computed quantum mechanically including the treatment of Cu surface vibration [81]. A standard model for solid state physics, a double well system coupled bilinear to a bath, was rigorously treated using the quantum transition state concept and (multi-layer) MCTDH calculations [83,84] by Wang, Thoss, and coworkers. This study extended previous work of Topaler and Makri [86] which relied on real time path integral calculations and offer a perspective to describe the real time quantum dynamics of condensed phase reactions beyond the limitation to harmonic bath models.

Efficient numerical techniques

In reaction rate calculations based on the quantum transition state concept, wave packets representing the vibrational states of the activated complex are started at the reaction barrier. They are then propagated towards the reactant and product channels. However, the propagation can be stopped as soon as all components of the wave packets have decided whether they will end up either as products or as reactants. For reactions showing significant potential barriers, this point of no return is typically reached after only a few ten femtoseconds. Thus, only short propagation times are required.

The multi-configurational time-dependent Hartree (MCTDH) approach [8,9] is ideally suited to facilitate numerically exact wave packet propagations under these circumstances. It then allows one to rigorously describe the quantum dynamics of polyatomic systems consisting of more than four or five atoms. The MCTDH approach employs a layered representation or the wavefunction

ψ(x 1 , .., x f , t) = n1 j1=1 .. nf jf =1 A j1..jf (t) f κ=1 φ (κ) jκ (x κ , t) , φ (κ) j (x κ , t) = Nκ l=1 c (κ) jl (t) • χ l (x κ ) . ( 4
)
The wavefunction is first represented in a basis of time-dependent expansion functions φ

(κ)
jκ (x κ , t) which are called single-particle functions. In a second layer of the representation, the single-particle functions are then represented in the basis of the underlying "primitive" time-independent basis functions χ l (x κ ). Standard discrete variable representation (DVR) [87][88][89] or fast Fourier transform (FFT) [90] schemes can be used to provide the χ l (x κ ). The equations of motion which describe the expansion coefficients A j1..jf (t) and c (κ) jl (t) of both layers can be obtained from the Dirac-Frenkel variational principle. It should be noted that extensions which can further increase the efficiency of the original MCTDH approach utilize the combination of several physical coordinates in a single "logical" coordinate (mode combination MCTDH [91,92]) and employ more than two layers in the wavefunction representation (multi-layer MCTDH [93][94][95]).

One further aspect of the MCTDH approach is particularly noteworthy in the context of reaction rate calculations for polyatomic systems. The equations of motion of the MCTDH approach show matrix elements of the potential energy V (x 1 , .., x f ) as, for example,

f κ=1 φ (κ) jκ V (x 1 , .., x f ) f κ=1 φ (κ) mκ . (5)
For reactive systems the potential energy functions V (x 1 , .., x f ) are typically involved functions of all coordinates and the correlation DVR (CDVR) [96] provided an efficient quadrature used in the MCTDH reaction rate calculations to evaluate these potential energy matrix elements. CDVR employs a layered time-dependent grid representation corresponding the MCTDH wavefunction: small sets of n κ grid points ξ corresponding to the N κ primitive time-independent basis functions χ l (x κ ) are used simultaneously. The combined use of the f grids (Ξ

(1) l1 , ξ (2) j2 , ξ (3) j3 .., ξ (f ) jf ), (ξ (1) j1 , Ξ (2) l2 , ξ (3) j3 , .., ξ (f ) jf ), .. , (ξ (1) j1 , ξ (2) j2 , ξ (3) j3 , .., Ξ (f )
jf ) and the grid (ξ

(1) j1 , ξ (2) j2 , .., ξ (f )
jf ) yields a efficient and accurate quadrature [94][95][96][97][98].

A second numerical technique, the rigorous statistical sampling approach [54,55,99], contributes to the efficency of reaction rate calculations which explicitly considered all J = 0 total angular momentum states [54,55,65,67] or higher levels of vibrational excitation present at high temperatures [70]. It basically is a scheme for the efficient evaluation of the trace. Consider a complete basis set {|χ l } of size N and an operator  represented in this basis,

A mn = χ m  χ n . (6)
Introducing M statistical functions

|φ j > = l (-1) αl(j) • |χ l > , j = 1, 2, .., M , (7)
where α l (j) denotes an integer random number, one finds that the average of their expectation values

A j = φ j  φ j = n,m
(-1) αm(j)+αn(j) A mn (8) converges in the limit of large M towards the trace of  lim

M →∞ 1 M M j=1 A j = n,m A mn   lim M →∞ 1 M M j=1 (-1) αm(j)+αn(j)   = n,m A mn δ nm = tr  . ( 9
)
The efficiency of the sampling scheme depends on the choice of the basis functions χ l . If A mn is diagonal, all A j are equal and only a single sample, i.e. M = 1, is required. More generally, the variance of trace computed using M sample depends only on the off-diagonal elements of A mn [54]. Thus, the sampling scheme is ideal for the description of weakly correlated degrees of freedom where approximate eigenfunctions of the operator in question, e.g. the Wigner rotation matrices for the description of rotational motion if the Hamiltonian is considered, are known. The scheme can favorably be combined with the MCTDH approach for the evaluation of flux correlations functions [54,55] and the calculation of partition functions [99].

Quantum effects in reaction rates: Conclusions

As already discussed in Sect.2.2, a large number of detailed theoretical studies of thermal rate constants was presented for diverse reactions within the last two decade. Based on these results, a consistent picture emerged which allows on to understand in detail diverse quantum effects on reaction rates as well as the relevance of the coupling of internal and rotational motion. These results are of particular interest considering the a larger community of theoretically interested chemists: understanding when computationally less demanding approaches and simplified models, standard classical harmonic transition theory in particular, can safely be employed is of crucial importance for the investigation larger molecular systems of applied chemical or biochemical interest. The present section therefore tries to concisely summarize central conclusions obtained from rigorous reaction rate calculations.

Quantum mechanical tunneling typically results in a significant increase of thermal rate constants of hydrogen transfer processes at temperatures below 500 K.

Comparing the results of accurate reaction rate calculations with results obtained by harmonic TST which do not account for quantum tunneling, quantum effects typically increase of k(T) by factors of about 5 to 10 at room temperature. The results for the H + CH 4 reaction displayed in Fig. 1 provide a typical example. Similar results were found for a number of other hydrogen-transferring reactions, e.g., [71]. While tunneling significantly enhances the reaction rates at temperatures below about 400 K, it tends to become less important above about 500 K and typically stops to be significant somewhere between 500 K and 1000 K. It should be noted that all these reactions show a rather imaginary normal mode frequency ω imag at the transition state which significantly exceeds the available thermal energy k B T : ω imag equals 1414 cm -1 for

HCl + OH → Cl + H 2 O [67] or O + CH 4 → OH + CH 3
H + CH 4 → H 2 + CH 3 [19], 1549 cm -1 for O + CH 4 → OH + CH 3 [73], 1192 cm -1 for H 2 + OH → H + H 2 O [100], or 1441 cm -1 for H 2 + Cl → H + HCl [101].
While quantum mechanical tunneling clearly is a prominent effect in reactions where hydrogen atoms are transferred, it is not particularly important for reactions where exclusively heavy atoms are involved. Despite opposite claims for processes as the dissociative adsorption of N 2 on a Ru-surface, quantum effects on the rate constants of such reactive processes are comparatively small [82]. As an example, Fig. 2 displays the thermal rate constant for the dissociative adsorption of N 2 on a stepped Ru-surface [82]. Here the enhancement due to quantum tunneling does not exceed 40 percent even at temperatures as low as 200 K. A simple Wigner tunneling correction [102]

k QM k T ST = 1 + 1 24 ω imag k B T 2 ( 10
)
is already sufficient to describe the quantum effects quite well, since here the imaginary frequency, ω imag = 409 cm -1 is much smaller compared to ω imag for hydrogen transfer reactions. Thus, the imaginary normal mode frequency at the transition state provides a good indication for the size of the tunneling effects to be expected. Studying a reaction where quantum effects are important, one might consider to avoid rigorous full-dimensional calculations and to employ a reduced-dimensional description to account for the quantum effects present. Several reduced-dimensional models were investigated for benchmark reactions as H + CH 4 → H 2 + CH 3 [78,[103][104][105][106][107][108][109] or [110,111] and compared to accurate fulldimensional results. Depending on the choice of the coordinates explicitly included and the modeling used to account for the neglected degrees of freedom, results of varying accuracy have been obtained. Approximate but full-dimensional transition state based theories tended to provide more reliable results for thermal rate constants [13][14][15][16]112]. Using well-adjusted reduced-dimensional models or transition state based approximate theories accounting for quantum effects, approximate reaction rate calculations rather predictably achieve agreements with accurate results within a factor of about two to three and in fortuitous cases even significantly better agreements were found. However, due to the intrinsic limitations of both types of approximate approaches, systematic improvement of such approximate results towards the accurate ones is generally not possible.

H 2 + OH → H + H 2 O
The above discussion about quantum effects in reaction rates has focused on tunneling. Zero point energy effects are, in principle, as least as important. Considering the H + CH 4 → H 2 + CH 3 as a typical example, the vibrationally adiabatic barrier height, which includes zero point energy in harmonic approximation, is 13.8 kcal/mol while the corresponding pure potential energy barrier is 14.9 kcal/mol [19]. Thus, zero point energy effects lower the effective barrier by ∆E 0 = -1.1 kcal/mol which corresponds to an increase of the room temperature rate of about a factor 7 (= e -∆E0/kBT ). However, since harmonic estimates for zero point energy effects are typically included even into the "classical" TST rate constants, this effect is usually not considered as a "quantum effect". While zero point energy effects can be easily included on the level of the harmonic approximation if the vibrational frequencies of the reactants and at the transition state geometry are known, a rigorous treatment accounting for anharmonic effects requires a more elaborate description. Since reactive and vibrational motion in the barrier region are usually strongly coupled, descriptions beyond the most simple harmonic approximation require truly multi-dimensional quantum dynamics calculations. Here it was found that the anharmonicities in principle have a sizeable effect on the zero point energies (e.g., around 0.5 kcal/mol for CH 4 [69,99] and the H +CH 4 transition state [70] at the Jordan-Gilbert PES [72]), but these effects cancel out quite well when the differences between the transition state energies and the reactant energies are considered [69,70]. Thus, approximate calculations can typically disregard such anharmonic effects as long as it is consistently done in the reactant partition function and the N (E) computation.

The effect of anharmonicities on the vibrational excitation energies present in the different vibrational contributions to N (E) [70,74] and the reactant partition function [99] were found to be irrelevant at the typical temperatures considered in quantum reaction rate calculations. Moreover, the contributions of the different vibrational states of the activated complex to k(T ) and N (E) were found to show approximately equal enhancement factors due to quantum tunneling [70,74]. Thus, calculating thermal rate constants as

k(T ) = Q ‡ rovib Q r 1 2π dE N GS (E) e -E k B T , (11)
where Q ‡ rovib is the rovibrational partition function at the transition state, Q r is the partition function of the reactants, and N GS (E) is the N (E) contribution of the ground vibrational state of the activated complex (i.e., the contribution of the pair of thermal flux eigenstates showing the largest absolute eigenvalues), typically provides an excellent approximation to the exact k(T ).

Isotope effects in hydrogen transfer reactions as H/D + CH 4 or H 2 /HD/D 2 + CH 3 were straightforwardly analyzed based on the above ideas [74,75]: primary isotope effects are strongly influenced by quantum mechanical tunneling since the mass of the transferring particle is varied. In contrast, secondary isotope effects are well described by standard harmonic TST: here only the zero point energies changes which is well described by the TST treatment as discussed above.

The above discussion focused on the internal degrees of freedom of the reacting system. A rigorous treatment has to consider overall rotation, i.e. rotation of the entire system of reacting molecules. In a TST description, overall rotation is generally considered to be separable from internal motion and a rigid rotor model is employed for the description of the rotational motion. Then the thermal rate constant can be calculated as

k T ST (T ) = Q ‡ rot Q ‡ vib 2πQ r k B T exp(-E b /k B T ) (12)
where E b is the (vibrationally adiabatic) barrier height and Q ‡ rot and Q ‡ vib are the rotational and vibrational partition functions of the transition state, respectively.

The separability of overall rotation and internal motion is frequently also assumed in otherwise rigorous quantum reaction rate calculations. This approach is called J-shifting [56]. Then k(T) is calculated as

k J-shif ting (T ) = Q ‡ rot Q ‡ vib 2πQ r dE N J=0 (E) e -E k B T (13)
where N J=0 (E) is the cumulative reaction probability computed for vanishing total angular momentum J. Since the J-shifting approach significantly reduces the numerical effort compared to calculations considering all J values explicitly, this approximation is highly attractive and its validity has been investigated in great detail for reactions as O + HCl → OH + Cl [45,47,54,55,62,113],

H 2 + Cl → H + HCl [59-61], H 2 + OH → H + H 2 O [64, 65], HCl + OH → Cl + H 2 O [67].
The accuracy of the J-shifting scheme was found to be directly be related to ability to define unique moments of inertia for the activated complex. If the activated complex is rigid and the moments of inertia vary only weakly over the dynamical relevant region, the J-shifting approach yields accurate results. Considering that for a nonlinear transition state the rotational partition is (in the classical limit) given by

Q = rot (T ) = π(kT ) 3 8I A I B I C , (14)
where I A , I B , and I C are the three principle moments of inertia of the activated complex, one can directly relate the error margin of the J-shifting approach to the variation of the moments of inertia during the time scale relevant for the determination of the reaction rate. This time scale is typically only in the range of a few ten femtoseconds. The J-shifting approach works particularly well for larger reactive systems since there all moments of inertia tend to be large and vary comparatively little in course of reaction.

Even if in cases where the transition state is only semirigid, the errors introduced be the J-shifting approach remain moderate: errors of 20 and 30 percent have found in the unfavorable cases as O + HCl [55] and H 2 + OH [65]. In these cases, more accurate results can be obtained employing the centrifugal sudden (CS) approximation [114][115][116]. Investigating reactions with linear transition states, the coordinate describing reaction around the linear axis must obviously be included in the explicit dynamical treatment. A J-shifting analogous procedure can then be employed to account for the two remaining rotational degrees of freedom [60] or one can resort to a CS description.

Potential energy surfaces for reaction rate calculations

Constructing multi-dimensional PESs

Considering the availability of accurate multi-dimensional PESs, the last decade witnessed tremendous progress. Until about ten years ago, accurate ab initio PESs were available only for selected three atom reactions. Meanwhile efficient techniques which allow one to construct truly multi-dimensional PESs based on a comparatively moderate amount of electronic structure calculations were developed.

The modified Shepard interpolation approach of Collins and coworkers [117][118][119][120][121][122][123][124] pioneered this development. In this approach, the potential energy surface (PES) is constructed as the weighted sum of local contributions:

V (q) = Nref i=1 w i (q) V local (q, q i ). ( 15
)
Here the local potentials V local (q, q i ) provide accurate descriptions of the potential in the vicinity of the reference points q i and the w i (q) are weighting functions guaranteeing a smooth interpolation of these local potentials. The local potentials are given as second order Taylor expansions and the required energies, gradients, and second derivatives (Hessians) at the reference points can be directly obtained from electronic structure calculations. Appropriate linear combinations of all inverse pairwise distances 1/r ij provide optimal coordinates to be used in the harmonic expansion. This use of specific curvilinear coordinates allows one to include a significant part of anharmonicities already on the level of the contributions V local of a single reference point. The quality of the interpolated PES depends on the position and number of reference points. By successively adding points, the quality of the PES can be improved until convergence (towards the limit given by the electronic structure theory employed). Combing the addition of new reference points with dynamical calculations which monitor the convergence and suggest the position of new reference points, a "semi-direct" approach interfacing electronic structure and dynamics calculations results.

Employing the Shepard interpolation approach, accurate PESs for the calculations of thermal rate constants of the four atom reaction [125][126][127] and the six atom reaction H + CH 4 → H 2 + CH 3 [18,19,74,75] were presented for the first time in 2000 and 2004, respectively.

H 2 + OH → H + H 2 O [
More recently, another efficient scheme for the construction of multi-dimensional PES was developed by Bowman, Braams, and coworkers [128,129]. It also employs dynamical calculations to suggest relevant geometries where electronic structure calculations are required, but then employs a fit to this data instead of an interpolation to construct the PES. The analytic form of the PES used in the fit is a polynomial depending on all the rescaled pairwise distances e -α•rij which form an over-complete set of coordinates. In the polynomials complete permutational symmetry of all identical atoms is enforced which drastically reduces the number of independent coefficients and thus allows one to use comparatively high order polynomials without requiring a prohibitively large amount of ab initio data.

The scheme was used to construct global PESs, e.g., for the H +CH 4 → H 2 +CH 3 [130,131] and F + CH 4 → HF + CH 3 [132] reactions. While this fitting based approach is very successful for the construction of global PESs requiring in scattering calculations, the Shepard interpolation approach has a distinct advantage when considering reaction rate calculations: interpolation avoids errors in the barrier height and the transition state vibrational frequencies. Thus, the most crucial quantity in a k(T) calculation, the vibrationally adiabatic barrier height, is accurately reproduced by the interpolation approach while the fitting approach here introduces additional errors.

PESs designed for reaction rate calculations

The thermal rate constant is determined by the dynamics in the vicinity of the reaction barrier. Thus, only the PES in this region really affects k(T). This fact simplifies the task of constructing an accurate PES to be used in reaction rate calculations. However, since the ratio of energy differences ∆E and the thermal k B T , very precise energy values are required. To accommodate these needs, a specific Shepardinterpolation based scheme for the construction of PESs specifically designed for reaction rate calculations was developed [133]. Application to the H + CH 4 → H 2 + CH 3 reaction [19] and the dissociative adsorption of N 2 on a stepped Ru surface found that only a very small number of reference points must be employed in the Shepard interpolation to achieve accurate results.

This finding is directly related to the physical nature of the process and should be discussed in the following in more detail. Fig. 1 shows the thermal rate constant computed for the H + CH 4 → H 2 + CH 3 reaction on a fully converged Shepardinterpolated PES (solid line) and an approximate PES which is only based on 23 reference points located at the minimum energy path (MEP) (dashed line). One finds that already the approximate MEP based PES yields quite accurate k(T) results. Consequently, only a small number of additional reference points is required to perfectly converge the relevant PES completely and less than 50 symmetry unique reference points were used in the fully converged k(T) calculation. An even more extreme result can be seen if a reaction showing smaller quantum effects is studied. Fig. 2 displays the thermal rate constant of the dissociative adsorption of N 2 on a stepped Ru surface. Here the full line shows results calculated on the converged PES while the symbols display results obtained for PESs based only on one or two reference points located close to the transition state. It is impressive to see that a single reference points is almost sufficient and that two reference points yield converged results. Considering the very nature of the transition state concept, these findings can be easily rationalized. Harmonic TST only requires the knowledge of the barrier height and the vibrational frequencies at the transition state: thus a PES based on a single reference point would be sufficient to compute k(T) within TST. The number of reference points required in the PES interpolation is therefore directly related to the extend of non-TST effects present in k(T).

Conclusions and perspectives

The present article reviewed the development in accurate reaction rate calculations over the last one to two decades. Concluding the review, two aspects should be pointed out in particular:

-Studying a variety of gas phase reactions which involve three to six atoms, a consistent picture explaining the extend and importance of quantum effects seen in thermal rate constants has emerged. Effects resulting from zero point energy, tunneling, occupation of vibrationally excited states of the activated complex, or the coupling of internal motion and overall rotation have been studied and understood in detail.

-Reaction rate calculations based on the quantum transition state concept, modern quantum dynamics approaches (MCTDH) and PES construction techniques as well as high level ab initio calculation allow one to accurately predict thermal rate constants for gas phase reactions involving -presently -up to six atoms. Considering recent developments in quantum dynamics as the multi-layer MCTDH approach [93][94][95] and the ability of the Shepard-interpolation based schemes to efficiently construct high-dimensional PESs specifically designed for the use in reaction rate calculations, accurate calculations for system consisting of even more atoms are in reach.

Moreover, just recently MCTDH calculations propagating wave packets from the transition state to the asymptotic region obtained initial state-selected reac-

  energy k B T enters into the rate constant roughly exponentially as e -∆E

  energies significantly more accurate than k B T which at room temperature takes values of k B T ≈ 1 mHartree ≈ 0.025 eV ≈ 0.6 kcal/mol ≈ 2 kJ/mol ≈ 200 cm -1 . While this clearly is a demanding task, these accuracies have been achieved in recent reaction rate calculations studying the H +CH 4 → H 2 +CH 3 reaction[18,19].

	February 8, 2011 15:16 Molecular Physics paper
	F o r
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  to efficiently perform the real and imaginary time wave packet propagations required Figure 1. Thermal rate constant k(T ) of the H + CH 4 → H 2 + CH 3 reaction [18, 19]: accurate quantum mechanical results and transition state theory results calculated on an accurate ab initio PES are displayed together with experimental results. Results obtained for a the PES which is based only on ab initio data from geometries located at the minimum energy path (MEP) are also shown.

	February 8, 2011 15:16 Molecular Physics paper
	6
		10 -15
		10 -16
		10 -17
	/s]
	3 k [cm	10 -18	exact QM
		10 -19	TST Marquire et al.
	Arai et al. Kurlyo et al. Back et al. Berlie et al. P o F 10 -20 2.0 2.5 10 -21 r MEP PES	3.0	3.5	4.0
			e
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  Figure2. Thermal rate constants k(T ) for the dissociative adsorption of N 2 on a stepped Ru surface[82,85]. The ratio k(T )/k T ST relative the transition state theory rate constant k T ST is displayed. Results from exact quantum mechanical calculations (full line) and the Wigner correction (dashed line) for the fully interpolated PES are given as well as quantum mechanical results calculated for PESs employing only one or two reference points in the modified Shepard interpolation.

	February 8, 2011 15:16 Molecular Physics paper
	10
	1.5
			exact QM
	1.4		1 point PES 2 point PES
			Wigner correction
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	r 200 300 1 o F 1.1	400	500	600	700
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  tion probabilities for the H + CH 4 → H 2 + CH 3[6,7]. Based on older ideas[43,134,135], these calculations utilized the quantum transition concept to calculate state-specific quantities observed in scattering experiments and provided a perspective towards an accurate and fully state-resolved description of polyatomic reactions as H + CH 4 , Cl + CH 4 , or F + CH 4 . Thus future studies might be able to combine transition state based ideas with state-selective scattering views to obtain an improved understanding of quantum state-specific chemistry.Financial support by the Deutsche Forschungsgemeinschaft, the Fond der Chemischen Industrie, and the European Commission through the RTN Program (HPRN-CT-1999-00007 and HPRN-CT-2002-00170) is gratefully acknowledged. The author particularly thanks Frank Matzkies, Fermín Huarte-Larrañaga, Tao Wu, Rob van Harrevelt, Gerd Schiffel, Gunnar Nyman, and W. H. Miller for many discussions and fuitful collaborations on the subject reviewed.

	February 8, 2011 15:16 Molecular Physics paper February 8, 2011 15:16 Molecular Physics paper
	5. Acknowledgments
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Harald Rösner
email: rosner@uni-muenster.de

Egor A Prokofyev

Ruslan Z Valiev
email: rzvaliev@mail.rb.ru

Gerhard Wilde
email: gwilde@uni-muenster.de

R Singh

S V Divinski

Annealing behaviour of nanocrystalline NiTi (50 at% Ni) alloy produced by high-pressure torsion

Keywords: crystallization, grain boundaries, grain growth user supplied): NiTi, Relaxation, Nanocrystalline NiTi, nanocrystalline, grain boundary, crystallization, relaxation

An equiatomic nanocrystalline NiTi alloy, deformed by high-pressure torsion (HPT) was investigated. The as-prepared bulk NiTi alloy consists of both amorphous and nanocrystalline phases. Crystallization and structural changes during annealing were investigated by differential scanning calorimetry (DSC), X-ray diffraction (XRD) and transmission electron microscopy (TEM).

The DSC thermograms and X-ray analyses reveal stress relaxation and partial crystallization below 500 K while grain growth of the nanocrystals occurs predominantly after heating to temperatures above 573 K. Along with the amorphous phase crystallization, a continuous growth of pre-existing nanocrystals that are retained after HPT was observed. The DSC signals observed during continuous heating experiments indicate an unusually large separation between the crystallization and the growth stages. A detailed analysis of the evolution of the enthalpy release upon annealing reveals reproducibly non-monotonous trends with annealing temperature that cannot be explained solely with nucleation and growth of crystalline volume fractions. Instead, the results can be rationalized by assuming a reverse amorphization process occuring during annealing at 523 K. This behavior, that also causes a large variation of the nanocrystals size after annealing at higher

Introduction

The intermetallic compound NiTi also known as "Nitinol" with near equiatomic composition of Ni and Ti has drawn much interest for its ability to be used, because of its unique combination of thermal shape memory effect, superelasticity and good damping [1][2][3][4][5]. Nitinol has many attractive service applications, particularly in the field of biomedical technology such as, dental archwires, bone staples, endoscopic instruments, stents and surgical retractors [6,[START_REF] Moberly | Engineering Aspects of Shape Memory Alloys[END_REF].

The development of nanocrystalline Nitinol has been the subject of intensive research, because fine grained or nanocrystalline material shows a variety of different and improved properties compared to the conventional coarse-grained polycrystalline material [START_REF] Birringer | [END_REF][9][10][11][12][13][14][15][16]. Recently, bulk nanocrystalline NiTi has attracted much interest as a high-tech material, showing enhanced shape memory behavior and superelasticity with ultra high recovery stress [9,10]. HPT (high pressure torsion) is one of the methods that allow obtaining nano-structured materials with new and unique properties and small grain size [11]. Combining HPT and subsequent annealing treatments one can tune the grain size and as a result the properties of the material can be controlled [12][13][14]. One prominent example is the size effect that can significantly shift the transformation from the cubic B2 high temperature phase to the monoclinic B19' martensite; grains with a size less than 50 nm can even lead to the complete suppression of the transformation [15,16].The kinetics of nucleation and grain growth was measured by various experimental methods, such as differential scanning [17][18][19][20], melt spinning (MS) [21][22][23][24] and cold rolling [START_REF] Dinda | Ultrafine Grained Materials III[END_REF][START_REF] Dinda | [END_REF].

Crystallization of the amorphous thin ribbons leads to microcrystalline structures [18,19]. Contrary to these results, crystallization at low temperatures of the NiTi alloy made amorphous by HPT leads to nanocrystalline structures [12, 13, and 27].

Sergueeva et al. has investigated the structural changes in a NiTi alloy produced by

HPT by in-situ heating in a transmission electron microscope (TEM) [12]. Very recently M. Peterlechner et al. also studied the crystallization of a HPT-processed

NiTi alloy [START_REF] Peterlechner | proceedings of the International Symposium on Bulk Nanostructured Materials[END_REF]. In the previous studies it has been reported by DSC and in-situ TEM that the crystallization of HPT deformed NiTi started at temperatures lower than 473 K and finished at 673 K and there was a continuous increase in the crystalline volume fraction during heating as well as annealing. The present study is aimed to analyze if the crystallization and the growth transformations are truly monotonous in temperature and to investigate the crystallization and the related structure changes in detail during annealing of amorphous NiTi alloy produced by HPT, using DSC, XRD and TEM.

Experimental details

Material

In the present study, a NiTi alloy with a composition of Ni discs cut from the quenched rod were subjected to HPT under a pressure of 5 GPa to a true logarithmic strain of 6, calculated by using the following formula:

ln r l θ ε   =     ( 1
)
Where ε is the logarithmic strain, θ is the rotation angle in radians, r is the radius and l is the thickness of the disc. The as-processed material is in the form of discs with diameters of 20 mm and thicknesses of about 0.8 mm. Discs of 4.5 mm (DSC & XRD measurements) and 3.0 mm (the TEM investigation) in diameter were cut from the asprocessed disc by the spark erosion technique. The distance between the centers of the HPT deformed disc was for the DSC & XRD 7.75 mm and for the TEM samples 8.5 mm, respectively. Hence, according to Eq. ( 1), the averaged logarithmic strain in the DSC, XRD and TEM samples is 5.72 and 5.81, respectively.

Characterization

Calorimetric measurements were performed by differential scanning calorimetry (Diamond DSC, Perkin-Elmer) under Ar atmosphere. A consecutive of DSC scans was recorded with the same specimen with a heating rate 10 K/min with a continuously increasing final temperature in the range from 453 K to 723 K. One hour annealing at the final temperature followed each of these DSC scans. In detail, the data were collected according to the following scheme:

(a) The as-prepared sample was heated to 453 K with a heating rate of 10 K/min followed by isothermal annealing for 1 h. After annealing, the sample was cooled down to 323 K using a cooling rate of 100 K/min.

(b) Immediately after the first run, the same sample was heated to 523 K, isothermally held at 523 K for 1 h, and subsequently cooled down to 323 K. (d) In this way, the same sample was heated to a consistently higher pre-selected temperature ranging from 453 K to 723 K followed by annealing for 1 h. The heating and cooling rates of 10 K/min and 100 K/min, respectively, were systematically used in each cycle.

In such a way, the thermally activated processes occurred during isothermal holding treatments could consistently be examined, avoiding uncertainties related to transient regimes during switching from heating to isothermal holding.

A second type of DSC scans was recorded to determine the heat release during annealing (δH) at temperatures ranging from 453 K to 623 K by using different samples. The following measurements were performed:

(1) One sample was continuously heated from 323 K to 723 K and the total heat release δH total was measured, Fig. 1a. The reproducibility of the signal was proven on different samples.

(2) Samples were continuously heated to the annealing temperature and the heat release during this heating (δH 1 ) was measured, Fig. 1b. The samples were annealed at different temperatures for 1h and then cooled down to 323 K.

(3) Finally the same samples used in the respective second step were heated from 323 K to 723 K and the heat release (δH 2 ) during this scan was measured, Fig. 1b.

For each sample, two identical DSC runs were carried out. The second run was used as a baseline, which was then subtracted from the first run to obtain the irreversible In order to monitor the microstructural evolution, X-ray diffraction was performed subsequently using a Siemens D5000 diffractometer with CuK α radiation (λ = 1.5406 Å) operating at 40 kV and 40 mA. These XRD spectra were recorded in the 2θ range from 25° to 90° with a scan speed of 0.2°/min and increments of 0.02°.

The TEM measurements were performed on as-prepared and thermally treated samples using a Libra FE200 and a Tecnai F20, transmission electron microscope.

Specimens for transmission electron microscopy (TEM) were prepared by electropolishing using a solution of HNO 3 (65%) and CH 3 OH in the ratio of 1:2 by volume. The electrolyte was cooled to -20 °C and the voltage applied for thinning was 10 V. Note that the selected area electron diffraction (SAED) images recorded on the Libra FE200 microscope contain artificial features due to the switching of the mechanical shutter when ultra-short exposure times were used for the CCD-camera. This is a machine-related problem and can not be completely avoided. However the artifacts do not corroborate the diffraction pattern and thus main message of SAED images is clearly seen and is not affected by these artifacts.

Results

TEM

The as-prepared samples consist of both a minority nanocrystalline and a majority amorphous phase after HPT, where the structure of the crystalline fraction These heating temperatures were chosen to analyze the microstructural evolution during the DSC scan shown in Fig. 1a. The samples are partially crystalline after heating to 453 K; also several amorphous bands are visible. On increasing the maximum temperature, the crystalline fraction is increasing and one can easily distinguish the crystalline and amorphous regions of the samples, as shown in the diffraction patterns of region A and B after heating to 563 K (Fig. 2d). There is no sign of grain growth and amorphous bands are clearly visible in the sample heated up to 563 K. Though the material appears to be completely crystalline after heating to 623 K, a narrow region of amorphous contrast still exists as shown in Fig. 2e. The diffraction pattern of this narrow region is not clearly amorphous, because the selected area aperture covered the crystalline area as well. After heating the sample to the maximum temperature of 723 K, the sample is completely crystalline and shows clearly visible grains with an average size of about 30 nm. This result shows that there is grain growth in the material at temperatures above 623 K.

A different set of samples was annealed in the DSC for 1 h at different temperatures in the range of 453 K to 723 K following heating with the rate of 10 K/min.

Afterwards, the microstructures of these samples were analyzed by TEM. Fig. 3a-b show that after annealing at 373 K and 453 K, the crystalline fraction is increased compared to the as-prepared specimen at RT (Fig. 2a) but still a significant amorphous fraction existed, as verified by the diffuse halo of the diffraction pattern.

The diffraction patterns obtained on the samples after annealing at even higher temperatures indicates that the sample is mostly crystalline after annealing at 523 K, although the bright field TEM image looks similar as the one obtained after annealing These results indicate that a small amorphous fraction is difficult to quantify from diffraction results. Further annealing at 723 K shows that the sample is not only completely crystalline but that there are clearly visible grains of an average size about 40 nm. A large variation in the grain size is observed after annealing at 723 K. Some grains have anomalously large sizes of about 120 nm.

DSC and XRD

A typical calorimetric signal measured on as-deformed nano-NiTi is shown in Fig. 1a.

Two exothermic processes can be distinguished; one starts at about 373 K and the second starts at 573 K with a peak heat release at 626 K (at a heating rate of 10 K/min), (Fig. 1a). A set of isothermal treatments (Fig. 4) always using the same specimen has been done by heating the sample with a constant heating rate of 10 K/min to sequentially higher temperatures ranging from 453 K to 723 K followed by isothermal holding for 1 h at each annealing temperature. There is an exothermic event on the signal during the first heating run to 453 K, where the area of this peak corresponds to the enthalpy δH I = 3.4 J/g. This exothermic peak is related to stress relaxation and the onset of the crystallization, as confirmed by the DSC scan in Fig. 4 and the TEM image shown in Fig. 2b.

The XRD spectra of as-prepared specimen (Fig. 5), indicate a shoulder peak at 2θ = 39.8° along with the peak corresponding to the partially amorphous phase at 2θ = 42.3°. The TEM examinations of Fig. 2 and Fig. 3 show the clear absence of the R-or B19´ phase. A layered structure consisting of neighboring bands of amorphous and partially crystalline phases is observed too, -a finding is consistent with the results of Peterlechner et al. [START_REF] Wibowo | [END_REF]. Based on these findings, we may suggest that the shoulder peak corresponds to a (deformed or strained) B2 NiTi phase appearing as separate bands in the microstructure. The corresponding difference in the lattice shoulder peak and the main peak, ∆a, is 0.1882 Å. The alternating thin layers of the B2 phase with different lattice parameters produce locally a bi-axial stress with a maximum value, σ, that can be estimated as:

a a E ∆ × - = ν σ 1 ( 2
)
where E is the Young modulus, ν is the Poisson ratio. Taking Young´s modulus of the NiTi alloy as 49 GPa [29], the maximum residual stress in the alloy is estimated as 4

GPa that is a reasonable value for a sample deformed at 5 GPa. The reduced heat required to relax this stress, U, is 1.38 J/g, calculated as:

ρ σ 1 2 1 × ∆ = a a U (3)
Here, ρ is the density of the NiTi alloy. This estimation fits reasonably the DSC measurement (the heat release is obtained as: δH I = 3.4 J/g, during heating to 453 K, Fig. 4). Therefore, we conclude that heating to 453 K results in a degeneracy of the shoulder peak and is probably related to the recovery of residual stresses in the HPTdeformed samples. By the observation of other DSC scans for heating to the temperatures between 523 K to 723 K, followed by annealing at each temperature for 1 h, it is clear (Fig. 4) that these curves are quite flat and the onset temperature of the exothermic signals are at 493 K, 553 K and 573 K, when the sample was heated to the maximum temperatures 523 K, 563 K and 623 K, respectively. At the higher temperatures no signal is measured after further annealing at 723 K. A large exothermic signal is observed on heating the specimen to 575 K.

A further calorimetric analysis has been done to explain the crystallization kinetics during isothermal annealing. Since the signal is rather weak, develops sluggishly and since the calorimetric sensitivity is proportional to the heating rate, direct determination during the isothermal annealing is not sufficiently accurate. To estimate the heat release δH during isothermal annealing, δH total , δH 1 and δH 2 were measured for the annealing treatment at different temperatures ranging from 453 K to 723 K and these values are shown in Fig. 6. δH is calculated in the following way:

δH = δH total -(δH 1 + δH 2 ) ( 4
)
δH total is the heat release during heating of the sample from 323 K to 723 K and has the constant value 38.62 ± 1.78 J/g. The heat release during continuous heating up to the annealing temperature, δH 1 , was measured and samples were annealed for 1 h at the final temperatures in the range from 453 K to 723 K. After that the same sample was continuously heated from 323 K to 723 K and the heat release during this heating, δH 2 , was measured. Thus the heat release during isothermal annealing, δH, is calculated using Eq. 4. The values of δH, δH 1 and δH 2 are plotted in Fig. 6 as a function of the isothermal annealing temperature. The data obtained from this analysis shows an unexpected but reproducible tendency: the value of δH decreases at T A = 523 K due to the increase in the value of δH 2 . This experimental result thus means that the crystalline fraction decreases with increasing temperature.

The XRD spectra after each annealing treatment in the DSC are shown in Fig. 5.

These XRD spectra were recorded on the same sample used for the DSC measurements (sample used for Fig. 4) during annealing in order to achieve maximum comparability between calorimetry and X-ray diffraction results. The as-prepared specimen reveals a disordered structure at room temperature with a very broad peak corresponding to the amorphous phase, centered at 2θ = 42.3° including the small peak at 2θ = 39.8°. Similar to the result obtained after annealing at 453 K a broad peak was observed at 2θ = 42°. This less broad peak indicates that on annealing at 453 of the amorphous phase. The spectra recorded after annealing at higher temperatures show that the material is mostly crystalline after annealing at 563 K while the sharpest peak observed after annealing at 723 K support the result obtained by the TEM and DSC observations. The peaks were identified as the (110), ( 200) peaks (211) of the austenite (B2) phase. To calculate the grain size of the NiTi alloy, Scherrer's formula [30] has been used, assuming that there is no strain in the material after annealing at 453 K,

{ } V K D cos λ β θ = (5)
and ( )

1 2 2 2 obs inst β β β = -
Where β is the integral width (in radians 2θ) of the reflection located at 2θ, β inst = 0.134 and β obs is the integral width calculated by XRD data (Fig. 5) at different temperatures, D v is the volume weighted crystallite size and K is the Scherrer constant with a value in the range of 0.87 -1.0. In the present study, K has been chosen as unity, λ is the wavelength of the radiation with a value of 1.54 Å. The grain sizes of the crystallites are listed in Table 1 together with the respective annealing temperatures.

Discussion

It is well known that NiTi is susceptible to amorphization as a result of various processing treatments and the formation of an amorphous phase was detected, for example in the case of ion beam mixing [31], mechanical alloying [32], ion implantation [33], shot peening [34], as well as cold rolling [24,[START_REF] Dinda | [END_REF]35,36] and shear deformation [37]. However in these publications the amorphization of NiTi was 13 observed on the surface or as discrete amorphous bands within a crystalline matrix. In this investigation similar to earlier studies [12] we demonstrate the formation of bulk amorphous NiTi alloy by HPT and the formation of a homogenous nanostructure by its crystallization and growth under controlled heating and annealing conditions. The as-deformed NiTi alloy was partially amorphous, with nanocrystals of extremely small sizes of 5-15 nm in diameter (Fig. 2a). This feature is also reported by Sergueeva et al. [12]. Fig. 2 shows the amorphous bands after heating to 453 K and 523 K, these bands are probably connected with deformation localization in the shear bands. The electron diffraction pattern made from an amorphous-nanocrystalline area (Fig. 2d) demonstrates the presence of the amorphous phase halo and extended reflexes from the nanocrystalline B2 phase. Peterlechner et al. [START_REF] Wibowo | [END_REF] have also reported the presence of well-defined interfaces of amorphous bands with respect to the nanocrystalline phase in the cross-section of HPT-deformed NiTi, while in case of the plane-section, the amorphous phase is obscured by overlapping nanograins in the TEM projection. In the present study, the amorphous bands in the plane-section are well-defined and straight interfaces with respect to the nanocrystalline phase (Fig. 2b).

The data analyzed in Fig. 6 shows an unexpected but reproducible tendency: the value of δH decreases at T A = 523 K due to the increase of the value of δH defects and a more disorders interface and are hence in an energetically more unstable state. Thus annealing at 523 K might trigger them to transform thermally into an amorphous state, as in this state half of the interface energy is saved as compared to the nanocrystalline state. The excess free energy densities due to the interface might have the effect of raising the free energy of the crystal to a level above that of the amorphous phase according to Koike et al. [35]. In addition, defects, especially dislocations, are transformed or completely annihilated upon amorphization consequently, due to the higher amorphous fraction after annealing at 523 K, there is an unusual increase in δH 2 (heat release during continuous heating from 323 K to 723

K after annealing at 523 K) and a corresponding decrease in δH. The value of δH at T A = 453 K is higher than the value obtained after annealing at T A = 563 K and 623 K due to the stress relaxation in addition to crystallization. It also implies that the annealing treatment induced crystallization at lower temperatures and grain growth at higher temperatures, this explanation is also supported by the TEM data shown in Fig. 3. The large variation in the grain size after annealing at 723 K, (Fig. 3d and3e)

shows that these grains are most probably those grains, which were already present after HPT deformation and became large during the annealing treatment. The results indicate that crystallization occurs by nucleation of new crystals and the rate of volume transformation due to nucleation is faster than the growth rate of pre-existing nanocrystals [38] at lower temperatures, while at higher temperatures grain growth of newly formed crystals and preexisting nanocrystals takes place simultaneously.

The grain sizes mentioned in Table 1, after annealing at 523 K and 723 K were used to estimate the decrease of the grain boundary energy after heating to temperatures above 523 K to 723 K (Fig. 7) by using the equation:

F o r P e e r R e v i e w O n l y 15 V A m H ⋅ ⋅ ∆ = ∆ ρ γ (7) Here, d V A ⋅ = 91 . 2
is the average surface area of the GBs in a polycrystal with a given grain size, d. Here ∆H/m is the heat release per unit weight, γ is the grain boundary energy, ∆A is change in the GB area, V is the volume and ρ is the density of the material. The calorimetric estimation of the grain boundary energy by Eq. 7 is 0.66 J/m 2 which is in good agreement with the grain boundary energy of 0.49 J/m 2 estimated in the literature by using Borisov´s analysis [39] in case of NiTi [40][41][42] for the same temperatures as used in the present study. The details of Borisov´s analysis are well explained in reference [43]. Although this value presents a rough estimate, it supports the interpretation that the main peak observed in the DSC scan (Fig. 1a) is related to the grain growth of the material. Thus the initial exothermic signal present in the DSC scan at temperatures below 573 K shown in "embryos" with sizes that are smaller than the critical radius for nucleation. Their presence could lower the energy barrier for forming a critical nucleus [44] and the corresponding activation energy of crystallization. This nanocrystalline debris along with nanocrystals is not found in the amorphous phase of NiTi obtained by SD or MS [38].

Summary

Homogenous nanocrystalline material with different grain size has been obtained by nanocrystallization of an amorphous phase using controlled annealing. Both the thermal stability and the crystallization kinetics of HPT-processed amorphous NiTi differ considerably from those reported in previous studies. One can tailor the properties of NiTi by obtaining the nanocrystalline structure, depending on the degree of the deformation and the temperature of the subsequent annealing treatments.

Additionally, the experimentally observed values of the enthalpy release and of the crystalline function as functions of the annealing temperature clearly show that they cannot be explained by a nucleation and growth process. In fact, the reproducible data obtained here cab be consistently explained by suggesting a reverse amorphization process in the HPT-deformed NiTi during annealing at 523 K for 1 h.

  ), X-ray diffraction and in situ heating in the transmission electron microscope (TEM) on thin amorphous ribbons of NiTi and NiTiCu made by sputter deposition (SD)

  Similar runs were repeated by heating to and holding at consequently higher temperatures.

  measurement signal and thus to remove the device function from the analysis.

8 of 10 K

 810 corresponds to the austenite (B2) phase, Fig 2a.The size of the nanocrystalline grains is extremely small with average diameters of about 5-15 nm. The samples were heated to the given maximum temperature in the range of 453 K to 723 K with a heating rate /min and cooled down with a cooling rate of 100 K/min. The corresponding microstructures that developed during these thermal treatments are shown in Fig.2.

  B2 NiTi phases corresponding to the angular difference between the

12 K

 12 the stress present in the NiTi alloy has relaxed but the material still consists mostly

2 .

 2 It is obvious, that this signature of the enthalpy release cannot be explained by nucleation and growth transformations, since these processes are monotonous in temperature. The possible cause of this unusual decrease in δH is a reverse amorphization during annealing at 523 K. The microstructure in Fig.2cshows amorphous bands between the crystalline regions. These amorphous bands are the region of deformation localization where the highest defect densities were created. The crystals which are boundaries of the crystalline and the amorphous regions have more

Fig. 4

 4 corresponds to the relaxation of the stress created by HPT and the crystallization of the material. The TEM image in Fig.2eshows the existence of a mainly crystalline microstructure along with a small amorphous fraction. It indicates that the most intense peak in the DSC scan in Fig.1ais related mainly to grain growth as well as the crystallization of the residual amorphous part. Thus the activation energy estimated in the present study is a convolution of the activation energy of grain growth and crystallization of the residual amorphous part. The activation energy of crystallization is expected to be low in the present case, since the temperature T P obtained in the present case of HPT deformed NiTi is almost 150 K lower than T P of crystallization measured on thin amorphous ribbons of NiTi obtained by melt spinning (MS) or sputter deposition (SD), with peak temperatures between about 773 K and 833 K[19,[21][22]. This result indicates a lower thermal stability of the amorphous phase obtained by HPT in the compared to the previously reported data on NiTi amorphized by different techniques. These differences might be caused by different structures of the amorphous phase in the as-deformed material: NiTi made amorphous by HPT contains nanocrystalline debris that can act as nucleation sites or nucleation

Figure 2 .

 2 Figure 2. Microstructure evolution of the NiTi alloy after heating to different temperatures.

Figure 3 .

 3 Figure 3. Microstructure evolution of the NiTi alloy after annealing for 1 h at different temperatures.

Figure 4 .

 4 Figure 4. The heat release by continuous heating of the NiTi alloy using the same sample to different maximum temperatures. The heating rate was always 10 K/min and the sample has subsequently been annealed for 1 h at each temperature.

Figure 5 .

 5 Figure 5. X-ray diffractograms showing the structure of the NiTi alloy before and after annealing at different temperatures for 1 h.

Figure 6 .

 6 Figure 6. Heat release vs. temperature (heating rate 10 K/min and annealed for 1 h at each temperature). Symbols are explained in the experimental details section.

Figure 7 .

 7 Figure 7. Heat release during continuous heating from 523 K to 723 K.

Figure 1 .

 1 Figure 1. DSC curve for continuous heating from 323 K to 723 K using a heating rate of 10 K/min, (a) ∆Htotal, and DSC signals step-wise annealing treatment (b) continuous heating from 323 K to 563 K is ∆H1, annealing for 1 h at 563 K followed by cooling down to 323 K and the same sample was continuously heating from 323 K to 723 K, ∆H2 . The DSC scan curve for ∆H2 is shifted for convenient comparison.

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .

 234567 Figure 2. Microstructure evolution of the NiTi alloy after heating to different temperatures.

  50.6 Ti 49.4 in atomic percent was used. This material was provided by the Institute of Physics of Advanced Materials, Ufa State Aviation TU, Russia. The initial rod produced by Intrinsic

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Devices Inc. (USA) with a diameter of 21 mm was water quenched from 1073 K. The

Table 1 .

 1 The large variability of the structural response of the material upon thermal or thermomechnical treatment explains the large range of reported structures of this class of alloys. At the same time, these results indicate the large potential for microstructure tailoring that is offered by the delicate balance between nucleation, grain growth and amorphization. Grain size of NiTi alloy after annealing at different temperature estimated by ∆H 2 . The DSC scan curve for ∆H 2 is shifted for convenient comparison.

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Acknowledgement

Financial support by the Deutsche Forschungsgemeinschft (DFG) is gratefully acknowledged. The authors also acknowledge the support from the Russian Foundation for Basic Research (project number 08-02-91955).