13428
J. Phys. Chem. 1996, 100, 13428-13432
Determination of End-to-End Distances in Oligomers by Pulsed EPR V. Pfannebecker, H. Klos, M. Hubrich,* T. Volkmer, A. Heuer, U. Wiesner, and H. W. Spiess Max-Planck-Institut fu¨ r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany ReceiVed: March 25, 1996; In Final Form: May 13, 1996X
Conformations of end-labeled alkanes in a rigid polymer matrix were investigated using the pulsed double electron-electron resonance (DEER) method. To this end, dipolar spectra of a series of aliphatic dicarbonic acids with 8-20 methylene units and chain ends bifunctionalized with nitroxide radicals were recorded in a polystyrene matrix. The biradicals are derived from hindered amine light stabilizers (HALS), used as polymer additives. Analysis of the dipolar spectra allows the determination of the end-to-end distances of the oligomer biradicals. The experiments show that the “flexible” alkane chains are almost completely stretched. This observation is consistent with the results of force field calculations.
a
Introduction The determination of intergroup distances in amorphous solids is of fundamental interest. For these materials the application of classical scattering techniques is limited because of the lack of periodic structures. One possibility to detect such distances is the analysis of the magnetic dipole-dipole interaction. For instance, the nuclear dipole-dipole couplings are successfully used in NMR spectroscopy to evaluate distances of atomic nuclei.1-7 However, the observable range by NMR is limited by the low magnitude of the nuclear magnetic moments.8 In contrast, EPR allows the determination of distances of dipolar coupled electrons up to several tens of nanometers, since the electron magnetic moment is about 3 orders of magnitude larger than that of the nuclei. Today a number of CW-(continuouswave) EPR methods are known to achieve this goal,9 where the analysis of the EPR line width is mainly used to investigate the electron-electron interaction. In crystalline and amorphous solids the EPR resonances are often inhomogeneously broadened, hiding smaller dipole-dipole interactions of distant electron spins. For these cases time-domain EPR methods were developed, which allow a direct measurement of the electronelectron interaction.10-14 Particularly the double-electronelectron resonance (DEER) experiment is designed to investigate the electron spin-spin interactions and their spatial distributions.15 Recently, a very detailed discussion of the DEER experiment was given by Larsen and Singel.16 The DEER experiment is based on excitation of two different spectral positions in the EPR spectrum. This can be achieved by two microwave sources of different frequencies and a double resonance cavity. At spectral position ω1 an electron spin echo (ESE) signal is detected, with fixed evolution time τ. During the evolution time an inversion microwave pulse at the second spectral position ω2 is irradiated and incremented in time. Due to the dipolar coupling of the electron spins an echo amplitude modulation evolves, from which the distance of the coupled electron-spins can be derived. In the present study, measurements of the end-to-end distances were performed for five different biradical molecules, in an amorphous matrix. As “rigid” standard a 2,6-bis[(((2,2,5,5tetramethyl-1-oxypyrolin-3-yl)carbonyl)oxy)]anthrachinon (BPA) biradical with fixed distance between two nitroxide radical groups was analyzed. Furthermore, four biradicals with flexible aliphatic dicarbonic acids with an increasing number n of * To whom the correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, July 15, 1996.
S0022-3654(96)00895-7 CCC: $12.00
b
Figure 1. (a) Bis[(((2,2,5,5-tetramethyl-1-oxypyrolin-3-yl)carbonyl)oxy)]-anthrachinon. (b) Flexible biradical molecules BRM(n) (n ) 8, 12, 14, 20) with an aliphatic dicarbonic acid end functionalized with nitroxide radicals.
methylene units (n ) 8, 12, 14, 20) and nitroxide radicals at the two ends of these chains were studied. These molecules can be considered as polymer model compounds. Experimental Section Sample Preparation. As “rigid” standard sample the biradical 2,6-bis[(((2,2,5,5-tetramethyl-1-oxypyrolin-3-yl)carbonyl)oxy)]anthrachinon (BPA, Figure 1a) was synthesized according to the route described by Larsen et al.16 Investigations of the rigid biradical in toluene were performed at a concentration of 2 mM. The structures of the flexible biradical molecules used in the present study are depicted in Figure 1b. They consist of aliphatic dicarbonic acids with a varying number n of methylene units (n ) 8, 12, 14, 20). The chain ends are bifunctionalized with nitroxide radicals. These biradicals or the unoxidized precursors were kindly supplied by Ciba Geigy, Basel. The structures are known as hindered amine light stabilizers (HALS) in industry and are used as polymer additives.17 In the following they will be denoted as BRM(n) (biradical molecule). Polystyrene was used as the polymer matrix in the measurements (Tg ) 378 K, Mw ) 218 000 g/mol) of the flexible BRM(n). Mixtures of the polymer and the biradicals were obtained by solvent casting using toluene as the solvent (c ) 2 mM/L). At low temperatures (T ) 80 K) polystyrene is frozen into the glassy state and can be considered as a completely rigid matrix.18 The concentration of the BRM(n) in polystyrene was 2 × 10-5 M/g in all samples. © 1996 American Chemical Society
End-to-End Distances in Oligomers
J. Phys. Chem., Vol. 100, No. 32, 1996 13429 error (∆ν/2 for the experimental frequency values has to be assumed for all of the measurements. Theory
Figure 2. DEER pulse sequence scheme. ω1 and ω2 are the microwave frequencies. The time t is the incrementation time, which is varied during the DEER experiment. τ is the fixed evolution time of the Hahn echo pulse sequence.
Experimental EPR Setup. Pulsed DEER measurements were performed on a modified Bruker ESP380E X-band Fourier transform spectrometer. One channel of the four microwave pulse forming units was used for a second microwave frequency. An HP-86290B rf plug-in unit in an HP-8350B sweep-oscillator (2.0-18.6 GHz) was employed as a source for the second frequency. The output microwave power of the rf plug-in is about 11 mW. Due to shifts of the resonator modes when the resonator is cooled down, it is necessary to control the absolute position of the frequency absorption of the microwave cavity at any temperature. Therefore, at the entrance of the DEER probe head a two-way switch (MDL) was installed to switch between the HP-microwave oscillator and an analyzing circuit. The latter consisted of a microwave circulator (RYT-Industries; Type 300033), a high-frequency Schottky diode (HP-423B), and an oscilloscope (GOULD 3150; 150 MHz). The home-built probe head contains a bridged-loop gap resonator (BLGR).19 For the design of the probe head special care had to be taken, because of the limited space inside the cryostat (Oxford Instruments Limited, CF935) and the necessity of high microwave pulse power. In order to save space, the probe head was constructed using a rectangular WR 62 Q-band waveguide of dimension 15.8 mm × 7.9 mm which is designed for microwave frequencies 12.4-18 GHz. For adjustment to X-band microwave the Q-band waveguide was filled with Teflon.20 A conic adapter outside the cryostat region connects the probe head with the X-band waveguide. Additionally, it is possible to turn the BLGR for adjustments and to couple the microwave with a polycarbonate (Makrolon, Ro¨hm, Darmstadt, Germany) iris by brass rods from outside. The employed BLGR exhibits two resonance modes, which shift in temperature. At room temperature and at T ) 80 K these two modes are separated from each other approximately by 10 MHz and by 60 MHz, respectively. The quality factor for both resonance modes is Q ≈ 1000. The DEER pulse sequence is shown in Figure 2. It consists of an Hahn echo pulse sequence with a fixed time delay τ between the pulses P0 and P1 at frequency ω1. A further π-pulse P2 is applied at resonance frequency ω2 at time t, which is incremented during the experiment. Within the Hahn pulse sequence the pulses have a correlated phase. However, between pulses at different frequencies ω1 and ω2 no phase correlation exist. Pulse lengths tp were chosen as tp(P0) ) 24 ns (π/2pulse, ω1), tp(P1) ) 48 ns (π-pulse, ω1), and tp(P2) ) 48 ns (π-pulse, ω2). This combination presents a compromise between selectivity of the spectral frequency excitation width and a favorable signal to noise ratio. A systematic discussion on the influence of spectral selectivity and microwave amplitude is given by Single and Larsen.16 The height of the echo-amplitude is determined by the transversal relaxation during the experiment, which limits the achievable resolution on the costs of signal to noise. The maximum evolution time τ ) 800 ns corresponds to a frequency resolution of ∆ν ) 1/τ ) 1.25 MHz. Therefore, an intrinsic
The theory of the DEER experiment is already well described.15,16,21 For the convenience of the reader, a concise description is given here, where we explain the DEER experiment in terms of an electron coherence transfer induced by the π-pulse P2. First we consider an isolated pair of electron spins, which is only coupled through dipole-dipole interaction.22,23 The electron spins are denoted with A and B. The Hamiltonian in the highfield approximation for such a weakly coupled two spin system is given by
H ) pωASZA + pωBSZB + pωABSZA SZB
(1)
where ωA and ωB are the resonance frequencies of the electron spin species A and B (SZA ) SZB ) 1/2), furthermore ωAB denotes the angular dependent coupling between the spin species, which is given by the well-known expression
ωAB ) ωdip(3 cos2 θAB - 1)
(2)
with the dipolar coupling frequency
ωdip )
|µBohr|2 gAgB 1 p r 3
(3)
AB
Here µBohr is the Bohr magneton and gΑ and gΒ are the effective g factors of spin species A and B, respectively. The vector rAB connecting the two spins is described by the distance rAB and its polar angle θAB with respect to the magnetic field B0. In the following we only consider the spin species A and its dipolar interaction with the spin species B. For this two-spin system, the A-spin transition is split into the two frequencies ωA+ ) ωA + 1/2ωAB and ωA- ) ωA - 1/2ωAB. The microwave carrier frequency ω1 is chosen such that the pulses P0 and P1 effect exclusively these two A-spin transitions, not irradiating the corresponding B-spin transitions. Thus, the (π/2)-pulse P0 transfers longitudinal A-spin magnetization into transversal magnetization, which evolves during the time t with frequencies ωA+ and ωA-, respectively. At the variable time t the π-pulse P2 at frequency ω2 flips the B spins, inducing a transfer of the electron coherences. Thus, the evolution frequency of the spin packet at frequency ωA+ is changed to ωA- (and vice versa), now evolving during the second time interval with length (τ - t). Finally, the π-pulse P1 at time τ induces a 180° phase flip of the evolving electron coherence. The phase φ of the A spins at time 2τ, where detection is performed, is now given by
φ(2τ) ) ωA(t + ωAm(τ - t) - ωAmt ) (ωABt
(4)
Since the quantum numbers of the B spins MSB are restricted to (1/2, there are two phase shifts with (ωABt relative to the magnetization My in the y-direction. Normalizing the echo amplitude I(t) at time t ) 0, we get
I(t) ) cos(ωABt)
(5)
Therefore, Fourier transform of this modulation function readily delivers the electron-electron coupling frequency ωAB. Due to the anisotropy of the dipolar coupling (see eq 2) the DEER spectrum is a superposition of coupling frequencies of radical spin pairs with distance r (Figure 1) of the radical pair
13430 J. Phys. Chem., Vol. 100, No. 32, 1996
Pfannebecker et al. TABLE 1: Conformations Matrix of the Dihedral Angles Determined by Monte Carlo Simulations of a Single Polyethylene Oligomer30 t q(j,k) ) g+ g-
Figure 3. EPR spectrum of the 14N-nitroxide BRM(20) in polystyrene. In the spectrum the excitation profiles at the microwave carrier frequencies ω1 and ω2 are shown. The difference of the two microwave frequencies is (ω2 - ω1)/2π ) 60 MHz. Adapted from ref 16.
centers, resulting in a Pake spectrum with its most prominent peak at ν⊥ ) ωdip/2π, corresponding to θAB ) 90°. Due to artifacts inherent in the DEER spectrum,15 caused by nonideal pulses, the reliable information given is the value of ν⊥ rather than the full frequency distribution. Hence in our quantitative analysis exclusively the value ν⊥ is used. The DEER signal not only contains contributions from coupled spins within the same biradical but also from dipolar interactions of all electron spins with each other. For amorphous systems this intermolecular part results in a decreasing exponential15,24
Iintermol(t) ) exp(-kCFBt)
(6)
with
k)
8π2|µBohr|2 gBgA 9x3p
(7)
where C denotes the concentration of unpaired electron spins and FB the relative part of the electron spins, excited by P2 at frequency ω2. Therefore, the time domain signal I(t) is described by a product of two functions: The intramolecular part Iintramol. of the radical spin pairs and the intermolecular part Iintermol, which leads to an exponential damping of the intramolecular signal
I(t) ) Iintramol(t) Iintermol(t) ) Iintramol(t) exp(-kCFBt) (8) Figure 3 shows a representative EPR spectrum of the rigid biradical in polystyrene. It resembles that of TEMPO in a rigid matrix. This indicates that the electron-electron interactions are weak. Furthermore, the two spectral excitation profiles of the microwave pulses (P0, P1, P2) with resonance frequencies ω1 and ω2 and their band widths of excitation are displayed. Corresponding to the irradiation bandwidth, nitroxides with certain orientations and specified nuclear quantum state mI are selected. Notice that the wings of the microwave pulse in the
t 0.5 0.66 0.66
g+ 0.25 0.27 0.07
g0.25 0.07 0.27
frequency domain should not overlap, since this introduces artifacts, by so-called “forbidden” electron-nuclear transitions, well-known from many electron spin echo envelope modulation experiments.25-28 Excitation of the spin ensemble with frequency ω1 is performed at the maximum absorption in the EPR spectra (see Figure 3), where the hyperfine quantum number of the nitrogen spin is mI ) 0. At frequency ω2, with a frequency offset of its microwave carrier frequency of 60 MHz, the pulse P2 is applied. Conformational Statistics of Polymers. The distributions of the end-to-end distances of the oligomer chains are determined by their conformational statistics.29 For polyethylene the minimum energy state is the all-trans state which corresponds to a fully stretched chain. In this limit we have φi ) 0° for all dihedral angles along the chain. For higher temperatures a few gauche states may occur as well. For polyethylene two different gauche states g( exist which correspond to φi ) (120° and one trans conformation t. Information about the probability of the occurrence of different conformations is readily available. Let p(j,k) denote the a priori probability that two adjacent dihedral angles correspond to states j and k (j,k ) t, g+, g-). This matrix has been determined from Monte Carlo simulations of a single oligomer by Raucci and Vacatello.30 From p(j,k) the matrix q(j,k) of conditional probabilities can be directly constructed which denotes the probability that the conformation is k, given that the neighbor conformation is j. The temperature, relevant for the present case, is the glass transition temperature of polystyrene (Tg ) 378 K) at which the conformations of the oligomers freeze. In what follows we use the conformational statistics as determined by Raucci and Vacatello for T ) 400 K. With q(j,k) ) p(j,k)/∑kp(j,k) one directly obtains from their data the probabilities listed in Table 1. Now one can successively generate the polyethylene chain. Let dihedral angle i be in state j. Then on the basis of a random number generator one can estimate the state of dihedral angle (i + 1) with the correct statistical weight. In this way polymer chains with arbitrary length can be created. From this ensemble of polymer chains we directly obtain the distribution of end-to-end distances. For comparison with experimental DEER data the size of the end-groups at both ends of the oligomers has to be taken into account when estimating the distance of the nitrogen atoms. For energetic reasons the radical tends to maximize its distance from the oligomer chain. Since the TEMPO group is rather flexible it is reasonable to assume that this adjustment only mildly influences the conformational statistics of the oligomer. Then, the size of the end group approximately corresponds to the length of six methylene units. Therefore, we attached the end groups to the oligomer such that the orientation vector d of the radical, connecting the final monomer of the oligomer and the nitrogen atom (|d| ) 6.8 Å), is parallel to the orientation vector of the final six monomers of the oligomer. In this way we calculate the distribution of the end-to-end distances of the nitrogen atoms. In the following section the maxima of the resulting spectra are compared with the maxima of the corresponding experimental spectra. Note that the conformational statistics is based on properties of a single chain. However, as
End-to-End Distances in Oligomers
Figure 4. DEER time signal of the rigid BPA in toluene at 80 K, with irradiation conditions as denoted in Figure 3. Pulse lengths: tp(P0) ) 24 ns, tp(P1) ) 48 ns, and tp(P2) ) 48 ns; constant evolution time τ ) 800 ns.
J. Phys. Chem., Vol. 100, No. 32, 1996 13431
Figure 6. DEER spectra of BRM(n) (n ) 20, 14, 12, 8) at 80 K as obtained by Fourier transform after baseline correction and zero filling.
TABLE 2: Overview of the Data Received from the DEER Spectraa
sample BRM(8) in PS(u) BRM(12) in PS(u) BRM(14) in PS(u) BRM(20) in PS(u)
calcd distance by conformation statistics exptl frequency exptl distance rtheory(ν⊥m)/Å ν⊥m/MHz r(ν⊥m)/Å 6.2 3.4 2.8 1.5
20.4 ( 0.7 24.8 ( 1.3 27.1 ( 1.9 32.8 ( 4.4
20.6 24 25.3 29.3
aν ⊥m denote the frequencies with highest amplitude in the dipolar spectra and r(ν⊥m) the corresponding distances. Model calculation with conformational statistics, see text.
Figure 5. Magnitude DEER spectrum of the rigid BPA in toluene at 80 K as obtained by Fourier transform of the time domain data in Figure 4 after baseline correction and zero filling.
is well-known from polymer physics, the static properties of a polymer chain in a glass strongly resemble those of a single chain.29,31 Results 1. Rigid Biradical Molecule. In order to optimize the experimental conditions and to have a standard of known endto-end distance, we first recorded DEER data for the rigid biradical BPA (Figure 1a). The DEER signal of BPA in toluene at 80 K is shown in Figure 4. The dominant amplitude modulation of the echo intensity corresponds to the intramolecular part, Iintra(t) whereas the damping results from the decaying exponential function and corresponds to the intermolecular part (eq 8) Iinter(t). With a least-squares fit the exponential decay was determined and removed from the original data. Fourier transform of the time-domain data (Figure 4) after baseline correction and zero filling yields the magnitude spectrum of BPA shown in Figure 5. Due to the experimental dead time, which is on the order of the pulse length, the spectrum is dominated by the most prominent peak of the dipolar spectrum, with ν⊥ ) 6.8 MHz. This yields a distance r ) 19.8 ( 0.7 Å between the two radical centers in agreement with molecular models as well as results obtained in an earlier investigation.16 2. Flexible Oligomers. End-to-end distance investigations were also performed on the four flexible oligomers BRM(n) with n ) 8, 12, 14, and 20 (see Figure 1b). The resulting DEER
Figure 7. End-to-end distances r vs number of methylene units in the polymer model compounds: (b) experimental, (- - -) theoretical (conformational statistics), (s) fully extended all-trans molecules (maximum limit of r).
spectra after correction for the exponential damping and Fourier transform of the time signal are shown in Figure 6. As expected, the frequency of the maximum decreases with increasing number n of methylene units, i.e., with increasing chain length. In Table 2 the frequency values ν⊥m of the maximum of the dipolar spectrum, together with the corresponding intramolecular distances of the radical centers, are compared with the distances calculated for the different compounds by the model described in the theory section. The results are shown in Figure 7, where measured and calculated values are plotted vs the number of CH2 units. The full straight line indicates the end-to-end distances of fully extended all-trans molecules and the dashed line describes the end-to-end distances of the biradical mol-
13432 J. Phys. Chem., Vol. 100, No. 32, 1996 ecules, calculated with the model described above. As expected, the end-to-end distances increase with increasing number of methylene groups n. Up to n ) 14 measured and calculated values of r are in remarkable agreement, indicating that the chains are stretched up to about 90% of the all-trans chain. The results presented here show that the DEER technique is indeed able to faithfully detect intergroup distances above 2 nm. As indicated by the error bars, the experimental value for n ) 20 has the highest uncertainty. This corresponds to the ν ∝ 1/r3 dependency, which causes larger error for longer distances since the corresponding frequency is relatively low. In addition, the baseline correction for the exponential damping can introduce systematic errors at such low frequencies. Theory predicts that with increasing chain length the end-to-end distance should deviate more and more from the all-trans distances, until finally the Gaussian statistics of linear polymer chains can be applied, for which the probability of end-to-end distance is given by29 P(r) ∝ exp(-3r2/(2Rg2), with the radius of gyration Rg. The experimental values of Figure 7 suggest that the oligomers are at least as high stretched as predicted from the conformational statistics. Further experimental work is in progress with enhanced spectral resolution to reduce the experimental uncertainty for longer distances. Conclusion and Outlook DEER experiments were performed on four flexible nitroxide end labeled polymer model compounds with varying number n of methylene units between the radical centers (n ) 8, 12, 14, 20). The dipolar spectra yield end-to-end distances of the molecules, which agree within the experimental errors with the distances calculated by conformational statistics. The results show that for n ) 8, 12, and 14 these flexible molecules are almost completely stretched with their radical center distances only approximately 10% shorter than those obtained for an alltrans conformation. For the largest model compound with n ) 20 a further deviation to shorter end-to-end distances is expected from theory. Better resolved spectra are necessary to verify this prediction. Furthermore, new pulse sequences are in development, where the DEER signal is refocused and deadtime effects are eliminated. This will yield full powder type spectra of electron-electron couplings. This study shows that meaningful values for intergroup distances as large as 3 nm can be determined in this way; this is of considerable interest for probing interfaces and cluster sizes in heterogeneous polymers and connectivities in site directed spin labeled biomolecules.32 The development of more elaborate experiments is therefore in progress. Acknowledgment. The authors thank C. Bauer for building the probe head and microwave resonator, R. Ulrich and A. Wagenknecht for synthesis of the BPA, Dr. C. Kro¨hnke (Ciba Geigy AG) for supplying the flexible biradicals and precursors, and J. Forrer (ETH Zurich) for helpful discussion. Financial
Pfannebecker et al. support from the Deutsche Forschungsgemeinschaft (LeibnizProgramm and SFB(262)) is gratefully acknowledged. Supporting Information Available: A picture of the experimental setup, with the modified Bruker ESP380E spectrometer (1 page). Ordering information is given on any current masthead page. References and Notes (1) Ernst, R. R.; Bodenhausen, G.; Wokaun, A. Principles of NMR in One and Two Dimensions; Clarendon Press: Oxford, UK, 1987. (2) Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and Polymers; Academic Press: San Diego, CA, 1994. (3) Sun, B.-Q.; Costa, P. R.; Kocisko, D.; Lansbury, P. T., Jr.; Griffin, R. G. J. Chem. Phys. 1995, 102, 702. (4) Schmidt, A.; Kowalewski, T.; Schaefer, J. Macromolecules 1993, 26, 1729. (5) Lee, P. L.; Kowalewski, T.; Poliks, M. D.; Schaefer, J. Macromolecules 1995, 28, 2476. (6) Lee, P. L.; Schaefer, J. Macromolecules 1995, 28, 1921. (7) Gottwald, J.; Demco, D. E.; Graf, R.; Spiess, H. W. Chem. Phys. Lett. 1995, 243, 314. (8) Powers, W. P.; Wasylishen, R. E. Ann. Rep. NMR Spectros. 1995, 23, 1. (9) Eaton,G. R.; Eaton, S. S. In Biological Magnetic Resonance, Berliner, L. J., Reuben, J., Eds.; Plenum: New York, 1989. (10) Rakowsky, M. H.; More, K. M.; Kulikov, A. V.; Eaton, G. R.; Eaton, S. S. J. Am. Chem. Soc. 1995, 117, 2049. (11) Kothe,G.; Weber, S.; Ohmes,E.; Thurnauer, M. C.; Norris, J. R. J. Phys. Chem. 1994, 98, 2706. (12) Kurshev, V. V.; Raitsimring, A. M.; Ichikawa, T. J. Phys. Chem. 1991, 95, 3564. (13) Raitsimring, A. M; Peisach,J.; Lee, H. C.; Chen, J. J. Phys. Chem. 1992, 96, 3526. (14) Kurshev, V. V.; Raitsimring, A. M.; Tsvetkov, Yu D. J. Magn. Reson. 1989, 81, 441. (15) Milov, A. D.; Ponomarev, A. B.; Tsvetkov, Yu. D. Chem. Phys. Lett. 1984, 110, 821. (16) Larsen, R. G.; Singel, D. J. J. Chem. Phys. 1993, 98, 5134. (17) Gugumus, F. Polym. Deg. Stab. 1993, 40, 167. (18) Elias, H.-G. Macromolecules; John Wiley & Sons; New York, 1977. (19) Pfenninger, S. Forrer, J.; Schweiger, A. ReV. Sci. Instrum. 1988, 59, 752. (20) Forrer, F. Private communication. (21) Tsvetkov, Yu. D. In Pulsed EPR: A new field of applications; Keijzers, C. P., Reijerse, E. J., Schmidt, J., Eds.; Koninklijke Nederlandse Akademie van Wetenschappen: Amsterdam, 1989; pp 206. (22) Kevan, L.; Kispert, L. D. Electron Spin Double Resonance Spectroscopy; John Wiley & Sons: New York, 1976. (23) Kevan, L.; Bowman, M. K. Modern Pulsed and Continuous-WaVe Electron Spin Resonance; John Wiley & Sons: New York, 1990. (24) Klauder, J. R.; Anderson, P. W. Phys. ReV. 1962, 125, 912. (25) Schweiger, A. Angew. Chem., Int. Ed. Engl. 1991, 30, 265. (26) Mims, W. B. Phys. ReV. 1972, B2, 2409. (27) Dikanov, S. A.; Tsvetkov, Yu. D. Electron Spin Echo EnVelope Moculation (ESEEM) Spectroscopy; CRC Press: Boca Raton, FL, 1992. (28) Hubrich, M.; Jeschke, G.; Schweiger, A. J. Chem. Phys. 1996. (29) Flory, P. J. Statistical Mechanics of Chain Molecules; WileyInterscience: New York, 1969; reprinted with the same title by Hanser, Mu¨nchen, 1989. (30) Raucci, R.; Vacatello, M. Makromol. Chem. Theory Simul. 1993, 2, 875. (31) Flory, P. J. J. Chem. Phys. 1949, 17, 303. (32) Altenbach, C.; Marti, T.; Khorana, H. G.; Hubbell, W. L. Science 1990, 248, 1088.
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