J. Phys. Chem. 1996, 100, 16419-16422
16419
A Stochastic Model for Crankshaft Transitions Giorgio J. Moro† Dipartimento di Chimica Fisica, UniVersita` di PadoVa, Via Loredan 2, 35131 PadoVa, Italy ReceiVed: May 17, 1996; In Final Form: July 16, 1996X
The model of three coupled rotors is analyzed by solving the corresponding Fokker-Plank equation of Smoluchowski type. From the numerical eigenvalues of the Fokker-Planck operator, the kinetic coefficients for conformational transitions are derived in the presence of bistable potentials for the torsional angles. Besides single-bond transitions through a saddle point of the internal potential, this procedure allows the identification of an anomalous type of kinetic processes, described as transitions of the inner rotor when the outer ones remain almost immobile (or rotate together). Their anomalous character derives from the absence of a saddle point crossing. Because of the localization within the chain, they can be assigned to transitions of crankshaft type. As expected on a qualitative ground, the increase of the friction of the outer rotors favors crankshaft transitions with respect to single-bond transitions. The implications of the model for the observation of crankshaft transitions in molecular systems are discussed in the conclusion.
Introduction A complete theory of conformational transitions in polymeric chains is still lacking because of the intrinsic complexity of the system. A major advancement toward this objective has been produced by the analysis of single-bond transitions due to Skolnick and Helfand.1 By applying the Kramers-Langer theory of saddle point crossing,2-4 they demonstrated that these transitions do not require large swinging movements of chain tails, because of the cooperative nature of polymer motions driving the kinetic event. The method supplies reasonable estimate of the transition rates,1,5,6 which can be employed also in the analysis of rotational motions of short alkyl chains.7 This theory, however, cannot explain all the features of conformational transitions in polymers. Several Brownian dynamics and molecular dynamics simulations8,9 have shown that independent single-bond transitions account for the majority of conformational transitions, but not all of them. A significant fraction of interconversion processes appear in the form of correlated bond transitions. Different explanations can be invoked. A picture suggested by Helfand is based on the chain stress induced by a torsional transition, which favors a subsequent torsional transition in adjacent positions.10 It is not easy to substantiate such a point of view in a formal theory, since it would require the explicit treatment of the dynamical coupling between the reactive coordinate for saddle point crossing and the collective librational degrees of freedom driving the equilibration within the intramolecular potential wells of the stable states. Only recently have there been efforts to analyze this dynamical coupling in simple model systems.11 An alternative explanation might be supplied by the so-called crankshaft transitions12 portrayed as nearly simultaneous transitions of two torsional angles which leave substantially immobile the chain tails (see the classifications in refs 13 and 14). Also within this framework, it is not evident how to derive a formal theory of the process. Only saddle points for single-bond transitions are recovered from internal potentials made of independent contributions of each torsional angle. Correspondingly, only these type of processes can be explained by applying the standard Kramers-Langer theory. † X
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[email protected]. Abstract published in AdVance ACS Abstracts, September 1, 1996.
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In a recent work concerning the relationship between kinetic equations and Fokker-Planck (FP) equations,15 an interesting result was reported for the model system of a two-bead chain bound to a wall in the presence of bistable potentials. Besides the ordinary transitions through a saddle point, an anomalous kinetic process clearly emerges when the friction of the outer bead is large enough. No saddle point can be associated with such a transition, and it can be portrayed in simple terms as the jump in position of the internal bead by leaving almost immobile the outer one. The analogy with crankshaft transitions was suggested by the absence of a saddle point crossing and by the localization of the transition inside the chain. In the present article the analogy is exploited further by studying the transition processes in a chain of three rotors. Certainly this is an extremely crude representation of a polymer chain, but on the other hand, it allows a simplified analysis of the effects of the viscous drag opposing the motion of chain tails, by imposing a large friction to the outer rotors. For this reason it can be employed as a conceptual model in order to investigate the dynamical processes leading to crankshaft transitions. The Model In Figure 1 the model chain of three rotors is represented as rods constrained to rotate about a common axis. In order to describe at the simplest level the orientational dynamics of the rotors, one can employ a Fokker-Planck equation of Smoluchowski type16,17 for the probability density p(R,t) of the orientations R ≡ (R1,R2,R3) of the three rotors with respect to a laboratory frame:
∂p(R,t) ∂t
3
)∑ j)1
kBT ∂
(
∂
+
ξj ∂Rj ∂Rj
1 ∂V kBT ∂Rj
)
p(R,t)
(1)
where V is the interaction potential among the rotors and ξj the rotational friction coefficient of the jth rod. A simple bistable potential will be used for each pair of adjacent rods
V ) (∆V/2)(2 - cos 2θ1 - cos 2θ2)
(2)
where ∆V is the torsional barrier height, while θ1 ≡ R1 - R2 and θ2 ≡ R2 - R3 are the torsional angles. Moreover, a © 1996 American Chemical Society
16420 J. Phys. Chem., Vol. 100, No. 40, 1996
Moro
Figure 1. The three-rotor system.
Figure 3. Kinetic scheme for the conformational transitions of the model system.
Figure 2. Contour plots from the potential function of eq 2. Circles, diamonds, and triangle represent minima, saddle points, and maximum, respectively. Dotted, continuous, and dashed lines denote equipotential lines with energy less than, equal to, and greater than ∆V, respectively.
symmetric chain will be considered by employing the same friction coefficient ξo ) ξ1 ) ξ3 for the outer rods. (ξi ) ξ2 will denote the friction of the inner rotor.) If information about the conformational dynamics only is required, the torsional angle dependence can be separated out after a simple change of variables18
∂p(θ1,θ2,t)/∂t ) -Γp(θ1,θ2,t) 2
Γ)-
∂
j
(
(
∂
∑ Dj,j′ ∂θ j,j′)1 ∂θ
j′
+
1 ∂V kBT ∂θj′
)
kBT/ξo + kBT/ξi -kBT/ξi D ) -k kBT/ξo + kBT/ξi BT/ξi
(3)
)
Standard numerical methods can be easily implemented in order to solve the FP equation and to calculate the eigenvalue spectrum of the time evolution operator Γ.19,20 A reduced representation is provided by kinetic equations for conformer populations, as suggested by the structure of the potential shown in Figure 2. The system has four equivalent minima at V ) 0 separated by saddle points with V ) ∆V. When the barrier height is much larger than the thermal energy, ∆V . kBT, the relaxation of conformer populations is well-separated in time scale from the small-amplitude motions about the potential minima.15 Correspondingly, a simple kinetic description of the conformational dynamics accounts for the long-time behavior of the system. Because of the symmetry of the model, only two independent kinetic coefficients are required, as shown in Figure 3 where opposite orientations of the rotors have been differentiated by means of arrows. Of course, only the relative orientations of the rotors are relevant when analyzing the torsional dynamics of the chain, the effects of conformational transitions on the overall orientation18,21 being not analyzed here.
The rate coefficient ws is assigned to transitions with π-jumps of one torsional angle only and which can be described by trajectories crossing a saddle points (see Figure 2). The kinetic scheme of Figure 3 allows also a second type of transitions with rate coefficient wc for π-jumps of both torsional angles. From the potential landscape of Figure 2 one recognizes that these processes cannot be assigned to normal transitions through a saddle point. A simple representation of them would be a straight trajectory between opposite minima through the potential maximum, i.e., the rotation of the internal rod when the outer ones are kept fixed (or rotate together). The analogy with crankshaft transitions should be evident if one considers the outer rotors as representative of chain tails remaining immobile during the transition of two torsional angles. On the basis of these ingredients, one can write the kinetic equations in the form of a master equation (ME)16 for the populations (concentrations) c ) (c1,c2,c3,c4) of the four conformers
(
dc(t)/dt ) -Wc(t)
)
2ws + wc -ws -wc -ws -ws 2ss + wc -ws -wc (4) W ) -w -ws 2ws + wc -ws c -ws -wc -ws 2ws + wc The relaxation rates are derived by calculating the eigenvalues of the transition rate matrix W λME j ME ME ME λME 0 ) 0, λ1 ) λ2 ) 2ws + 2wc, λ3 ) 4ws
(5)
Only two of them are independent because of the degeneracy ME ME λME 1 ) λ2 and the vanishing of λ0 for the stationary solution. A complementary set of kinetic relaxation rates can be obtained from the eigenvalues of FP operator Γ of eq 3 FP FP FP λFP 0 ) 0, λ1 ) λ2 , λ3
(6)
which for ∆V/kBT . 1 are well-separated from the remaining FP eigenvalues λFP 4 , λ5 , ... describing the equilibration about the potential minima.22 Also in this case only two kinetic rates are independent. A general procedure for the derivation of kinetic coefficients from the solutions of the FP equation in multistable problems has been presented in ref 15. In the present case, however, one can simply exploit the identity of the two sets of ) λFP eigenvalues, λME j j for j ) 0, 1, 2, 3, that is, the constraint that the kinetic equations should describe the same long-time behavior of the complete solutions of the FP operator. This identity can be resolved with respect to the transition rates FP FP ws ) λFP 3 /4, wc ) λ1 /2 - λ3 /4
(7)
and in this way one can derive the exact values of the two
A Stochastic Model for Crankshaft Transitions
Figure 4. Rates ws of transitions through saddle points (circles) and wc of crankshaft transitions (triangles) calculated for ∆V/kBT ) 5.
independent kinetic coefficients from the numerical eigenvalues of the FP operator. The results for the transition rates are shown in Figure 4 as a function of the friction coefficients for ∆V ) 5 kBT, which corresponds to the typical barrier of trans-to-gauche transitions in alkyl chains at room temperature. Because of the scaling of the rates according to the diffusion coefficient kBT/ξi of the inner rotor, these results depend only on the friction ratio ξo/ξi. As a convenient way to analyze these data, one can imagine the system with increasing friction ξo of the outer rod, while the friction ξi of the inner rotor is kept constant. The lowering of the rate ws of single-bond transitions is easily understood by taking into account that the saddle-point crossing always requires the rotation of the outer rods. A quantitative estimate can be derived from the Kramers-Langer theory of saddle-point crossing. On the contrary, the rate wc of crankshaft transitions is weakly dependent on ξo (a change by a factor of 2 for an increase of ξo by 2 orders of magnitude), as suggested by the simple picture of these transitions as the jump of the inner rotor alone. The same picture leads to unfavorable energetics with respect to single-bond transitions. In fact, wc is much smaller than ws when the two friction coefficients are comparable, and in this situation the conformational kinetics is substantially described by single-bond transitions alone. However, the increase of the friction of the outer rotors necessarily penalizes the single-bond transitions, and for ξo = ξi/10 the two rates becomes comparable. In such a case the unfavorable energetics of crankshaft transitions is counterbalanced by the low friction opposing it. Larger friction ratios lead to a predominance of crankshaft transitions. Some information about the system evolution during the crankshaft transitions can be derived from the shapes of the FP kinetic eigenfunctions. This type of analysis performed in ref 15 for an analogous system leads to the conclusion that the potential maximum is avoided by means of small displacements of the element with higher friction. The examination of the activation energies of conformational transitions in the threerotor model supports this conclusion. Let us consider the system with a fixed barrier height (times the Avogadro constant) of 12.4 kJ/mol and constant diffusion coefficients Di ) kBT/ξi, and Do ) kBT/ξo in the ratio Di/Do ) ξo/ξi ) 10. In Figure 5 the transition rates are represented as function of the temperature in the form of an Arrhenius plot. Given the almost linear profiles, one can analyze the temperature dependence of the rates according to the phenomenological Arrhenius rule, ws ∝ exp(-∆Es/RT) and wc ∝ exp(-∆Ec/RT), so obtaining the activation energies ∆Es ) 9 ( 1 kJ/mol and ∆Ec ) 13.4 ( 0.3 kJ/mol for the two types of transitions. ∆Es is smaller than the barrier height because of finite barrier effects and the change with the temperature of the preexponential factor of the rate.3,4 It should be emphasized that the activation energy ∆Ec for crankshaft transitions is considerably smaller that the energy
J. Phys. Chem., Vol. 100, No. 40, 1996 16421
Figure 5. Arrhenius plot for the transition rates (same symbols of Figure 4) calculated with a barrier height of 12.4 kJ/mol and constant diffusion coefficients in the ratio Di/Do ) 10.
difference 24.8 kJ/mol between potential maximum and minima. Because of finite barrier effects like for ∆Es, one cannot identify ∆Ec exactly with the highest potential energy acquired during the transition. Still the magnitude of ∆Ec suggests that the system performs crankshaft transitions without passing through the potential maximum. In the diagram of Figure 2, one should imagine a curved path connecting two opposite minima, so avoiding the potential maximum. (This requires small rotations of the outer rods.) If the two torsional angles are observed separately, such a kinetic event would then appear as a sequence of two single-bond transitions shortly separated in time, like the correlated bond transitions found in the simulations of polymers.8,9 Conclusion The major benefit deriving from the analysis of the threerotor model is the precise characterization of crankshaft transitions which have been proposed several years ago on a purely phenomenological ground. The simplicity of the model and its intrinsic symmetry allows the numerical determination of the rate of crankshaft transitions, with the opportunity of investigating the effects of specific molecular features. It has been shown that a large friction of the outer rotors, which might be considered as representative of chain tails, favors the crankshaft transitions with respect to single-bond transitions. Hopefully, the simplicity of the model might facilitate the derivation of an asymptotic theory like the Kramers-Langer theory of saddle-point crossing, also for crankshaft transitions. Still it is impossible to give a precise answer to the question concerning the presence of crankshaft transitions in polymer molecules in solutions. The behavior found in the three-rotor model seems to be compatible with the observation of correlated bond transitions found in molecular dynamics simulations. However, in order to verify this conjecture, the rates of crankshaft transitions must be compared with the rates of singlebond transitions in realistic stochastic models of threedimensional chains. One cannot employ the same method of this article because of the difficulties in the full numerical solution of the FP equation with a large number of degrees of freedom. This objective could be reached only if an asymptotic theory of crankshaft transitions would be available. The issue about the weight of crankshaft transitions with respect to single-bond transitions can be examined from another point of view. Instead of trying to give an answer about the presence of crankshaft transitions in a polymer, one could select specific molecular systems where these transitions have to be important as suggested by the analogy with the three-rotor model. The simplest analog is the terphenyl system represented in Figure 6. Let us assume that its internal potential is simply the superposition of the torsional potentials of biphenyl, whose
16422 J. Phys. Chem., Vol. 100, No. 40, 1996
Moro References and Notes
Figure 6. A candidate for the observation of crankshaft transitions.
general features are well-known:23 minima for the torsional angle near (45° and (135° and maxima in the planar and orthogonal configurations. If the transitions through planar configurations are neglected because of excluded-volume effects, a simple kinetic scheme like that of Figure 3 can be employed. By increasing the rotational friction of the outer rings, i.e., by inserting a bulky substituent for R of Figure 6, crankshaft transitions become favored with respect to saddle point crossings, as shown by the theoretical results of Figure 4. On the basis of the similarity with the three-rotor model, one can figure out other molecular systems for observing crankshaft transitions. Also, the main-chain polymeric liquid crystals, whose relaxation properties have been intensively studied by means of magnetic resonance techniques,24 might be seen from this point of view. In these polymers large and rigid mesogenic units are linked by flexible spacers like alkyl chains. The analogy with the threerotor model should be evident if one consider the prototype system with only one spacer connecting two bulky mesogenic units. Of course, the internal dynamics of the spacer is more complicated than that of the three-rotor model. Still one expects that the large friction of the mesogenic units should emphasize the role of crankshaft transitions, particularly in systems with short spacers. Acknowledgment. This work was supported by the Italian Ministry for Universities and Scientific and Technological Research and in part by the National Research Council (CNR) through its Centro Studi sugli Stati Molecolari and the Committee for Information Science and Technology.
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