NANO LETTERS
A Tunable Dendritic Molecular Actuator Paul M. Welch*
2005 Vol. 5, No. 7 1279-1283
Theoretical DiVision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Received March 4, 2005; Revised Manuscript Received May 12, 2005
ABSTRACT I present an electroresponsive molecular actuator based upon a diblock copolymer of a positively charged dendrimer and a negatively charged linear chain. Brownian dynamics simulations demonstrate the hybrid polyampholyte’s ability to apply a force or assume an equilibrium molecular strain tunable with an applied electric field. The free energy as a function of molecular strain at differing electric field strengths, as obtained via the Jarzynski identity, suggests a phase transition in the hybrid.
Molecular devices for applying strains or forces on demand play a central role in many envisioned nanomachines. Thus, there are a number of such actuators proposed in the literature.1 Many rely upon local conformational changes in their chemical constituents that translate into relatively small but rapidly applied strains or are best described as two-state rotors. Typical polymeric actuators display percent strains in the 3 to 30% range. If we are to mimic biological processes, however, large strains will also be of interest. For example, the actomyosin complex produces displacements in the range of 5 to 10 nm.2 Global conformational changes in linear polymers offer one route to achieve such largescale motions. Electrostatic attraction seems an obvious choice to effect these conformational changes since the force generated can be quite high and the presence of charges permits coupling of the device to an external electric field. In this letter, I present a construction designed to mimic a spring-loaded reel, with a linear chain serving as the molecular analogue of a string that is pulled onto a dendritic molecular spool. Electrostatic attraction serves as the spring while an applied electric field allows the compliance to be tuned. Several experimental3 and theoretical4 studies demonstrate that linear chains will complex with dendrimers, especially if they are oppositely charged. This can result in a dramatic change in the dimensions of both species. A number of investigators5 have also synthesized and characterized hybrids with linear and dendritic blocks in copolymers. Gitsov and Fre´chet6 found that linear-poly(ethylene oxide)-co-dendriticpoly(benzyl ether) exhibited hydrodynamic behavior consistent with a conformational arrangement of the linear block around the dendrimer when the hybrid was placed in a poor solvent for the branched block. Similarly, Mackay and coworkers7 found that hybrids display amphiphilic behavior * E-mail:
[email protected]. 10.1021/nl050422c CCC: $30.25 Published on Web 06/22/2005
© 2005 American Chemical Society
as a function of molecular weight in a variety of solvent conditions. Mackay8 further suggests that this conformational change may serve as a basis for a nanomotor. The device proposed here, illustrated in Figure 1, utilizes this hybrid topology and charge-induced complexation to wind a linear chain onto a spindle provided by a dendrimer. The construction consists of a sixth generation monocentric dendrimer, such as monocentric dendritic-poly(propylenimine), with charges at every branch point and terminal group. A noncharged, short tether chain is grown from one of the dendrimer’s terminal groups and is immobilized at the chain end on a hard-wall surface. From another terminal group, topologically opposed to the growth point of the tether, another linear chain is attached. This chain is oppositely charged to the dendrimer and has the same number of charged groups as the dendrimer so that the overall system is charge-neutral. Brownian dynamics simulations9 were applied to study a model of this device. The Debye-Hu¨ckel potential, UC/kBT ) lBqRqβ exp[-κr]/r, approximates the interaction between two unit charges qR and qβ separated by a distance r. Each bead in the dendrimer has charge qD and each segment of the charged chain carries qC such that qD ) -qC ≡ q. Counterions and additional solution salt are considered through the Debye length, κ-1, which is inversely proportional to the square root of the salt concentration. Reported lengths are in units of the Bjerrum length lB, roughly 7 Å in water at 25 °C.10 All of the beads interact via a LennardJones potential, ULJ) {(σ/r)12 - 2(σ/r)6}, parametrized such that the hybrid is in a good solvent at the simulated temperature, kBT/ ) 5, with σ/lB ) 0.8. The bonds are described by a harmonic potential, UB ) k(l - l0)2, where l and l0 are the bond length and the equilibrium bond length, respectively. To prevent bond crossing, l0/lB ) 0.7 and a maximum of l/lB ) 1.0 was enforced. The spring constant klB2/ ) 1000 maintained stiff bonds.
Figure 1. The model molecular actuator. The hybrid polyampholyte, composed of a charged generation 6 dendrimer and a linear chain of equal molecular weight and equal but opposite charge, is tethered by a noncharged leash to the hard wall. The electric field is normal to the surface.
An electric field E was applied normal to the surface and acts to pull the chain away from the surface and the dendrimer toward it, except in the one instance noted below in which the field polarity was reversed. This contributes an additional force of magnitude |Eq| to each of the charges along the axis normal to the surface. The details of the simulation algorithm are identical to those outlined in reference 11. The equilibrium free energy FE as a function of the distance X from the chain end to the center of the dendrimer was determined via the Jarzynski12 identity. To accomplish this, a phantom spring was attached to the free chain end and retracted at a constant rate dz/dt) 0.1/lB normal to the surface, where t is the dimensionless time in the simulation. This spring, mimicking an atomic force microscope’s cantilever, was modeled with a harmonic potential Up ) kp(lp - lp0)2 with kplB 2/ ) 0.1 and lp0/lB ) 0.1, where lp is the length of the spring. The starting configuration was obtained by first equilibrating the hybrid under the electric field strength of interest. The procedure delineated by Hummer and Szabo13 for applying the Jarzynski identity permits the reconstruction of the equilibrium free energies shown in Figure 2 for the case of a high solution salt concentration (approximately 1 molar if the solvent were water at room temperature) with (lBκ)-1 ) 0.42. The approach 1280
Figure 2. Estimates for the free energy FE as a function of the distance X from the center of the dendrimer to the end of the linear charged chain. Data for a range of field strengths (EqlB/ ) 0 to -1) and high salt concentration ((lBκ)-1 ) 0.42) are shown. The snapshots are taken from near the indicated minima.
requires that numerous “pulling” simulations be performed and that a histogram of the forces sampled as a function of the displacement be used to estimate the free energy. Ten independent trajectories were computed for each curve in Figure 2 to provide an estimate of the free energy. The details of how to perform this extraction are succinctly presented in ref 13. The free energy appears to be a simple quadratic near its minimum but stiffens at higher values of X. The left portions of the wells, corresponding to having the dendrimer and chain end closer than their equilibrium arrangement, were obtained by choosing conformations with X slightly less than the equilibrium value Xe as the starting point for the extension simulations. The field strength was increased at a constant interval, yet the minima of the curves shift in a nonuniform fashion. As carefully examined by Gore, Ritort, and Bustamante,14 the Jarzynski free energy estimator provides a slightly biased approximation to the true underlying free energy difference between two states. Specifically, they note that the mean square error falls off inversely with the number Nano Lett., Vol. 5, No. 7, 2005
Figure 3. The equilibrium distance Xe between the center of the dendrimer and the end of the charged chain as a function of the applied field E. Data for (lBκ)-1 ) 0.42 are shown.
of trajectories collected, raised to an exponent between zero and unity. Specifying this exponent for a specific system is not straightforward, unfortunately. However, increasing the number of trajectories from 10 to 90 typically changed the magnitude of the minimum by less than a 0.2 kBT, suggesting that the error is relatively small in these measurements.15 While more careful measurements are merited for specific chemical designs, the goal here is only to elucidate the general trend and especially the location of the minima; the latter showed no sensitivity to the number of trajectories collected. Figure 3 illustrates the location of the minima as a function of applied field strength for this same salt concentration. The equilibrium displacement Xe remains constant until a critical field strength E* is applied, after which there is a rapid rise in the value of Xe. Qualitatively, this indicates that the hybrid undergoes a phase transition similar to that of an Ising magnet in an applied field since the susceptibility dXe/dE appears to diverge at the critical field strength E*. The displacement Xe serves as the order parameter analogous to the magnetization in the Ising magnet, and there is a one-to-one correspondence between the applied fields in both models. Immediately after the critical electric field has been applied, Xe rises rapidly but quickly transforms into a simple power-law relation, which itself must ultimately break down at much higher fields when the chain becomes fully extended. The critical field strength E* is expected to shift to higher values with decreasing solution salt concentration; the less salt in the system, the more tightly bound is the chain to the dendrimer and, thus, more energy is required to separate them. Figure 4 depicts E*, determined from equilibrating an initially fully extended chain under a range of field strengths, versus κ-1. The transition becomes much sharper at lower salt concentrations such that each estimate is easily bounded. The solid line represents the best fit to an exponential of κ while the points are the estimates of E* bounded by the error bars given. Surprisingly, the literature contains no theoretical prediction for the field strength required to separate two oppositely charged linear chains, presumably a simpler problem than that studied here. Nano Lett., Vol. 5, No. 7, 2005
Figure 4. The estimated critical field strength E* required to separate the chain from the dendrimer as a function of Debye length κ-1. The error bars indicate the bounds for the estimate while the solid line indicates a best fit to an exponential.
Figure 5. Force-extension profiles under differing field strengths and Debye lengths. Ten trajectories are superimposed for each condition. The black symbols represent the retractive force F for the conditions EqlB/ ) 0 and (lBκ)-1 ) 0.42, the red correspond to EqlB/ ) -1 and (lBκ)-1 ) 0.42, the green indicate the results for EqlB/ ) 0 and (lBκ)-1 ) 1, and the blue are for EqlB/ ) -1.9 and (lBκ)-1 ) 1. The displacements X are normalized by the maximum elongation, Xmax.
However, the exponential seems an obvious choice since the critical field strength must ultimately plateau at zero added salt. In addition to the tunable equilibrium displacement just discussed, the hybrids also exert a retractive force F that can be controlled by the applied field strength. In the absence of an applied field, the force-elongation profiles for hybrids in a high salt solution resemble that of a simple, noncharged linear chain; initially the retractive force follows a simple Hookean law, while at higher elongations a nonlinear retractive force is operative, qualitatively similar to the Langevin force-extension profiles observed for linear chains.16 This is illustrated in Figure 5 by the black symbols for zero field and (lBκ)-1 ) 0.42. Applying an external field translates the curve to a higher elongation while limiting the range of the initial Hookean regime, as shown by the red symbols in Figure 5 where EqlB/ ) -1 and (lBκ)-1 ) 0.42. Lowering the salt concentration and pulling the chain away from the dendritic block in a field below the critical field strength results in a dramatic change in behavior, as shown by the 1281
green symbols in Figure 5 for EqlB/ ) 0 and (lBκ)-1 ) 1. Initially the force rises rapidly but then grows more slowly as more segments are removed from the dendrimer, similar to the behavior noted by Chaˆtellier and co-workers17 in removing a polyelectrolyte from an oppositely charged surface. The change in slope results from removing the segments further than the Debye length; unlike Chaˆtellier’s results for an unbound chain, the slope does not drop to zero since the chain is being stretched. The blue symbols in Figure 5, representing EqlB/ ) -1.9 and (lBκ)-1 ) 1, show that applying an external field in these lower salt conditions again shifts the curve, and the Langevin-like force-extension profile is recovered. Assuming that the dendritic and linear portions of the device are of commensurate size permits an estimate for the upper limit on the percent strain that may be achieved by such a device. The model presented here sets the step length of the linear portion equal to the step length between branch points in the dendritic block. The approximate distance between branch points in synthetic dendrimers is of the order 5 Å. Thus, the contour length of the linear block is roughly 945 Å. Similarly, the tether and dendritic block contribute an additional 85 Å to the fully stretched device. If the chain is completely encapsulated by the dendritc block at equilibrium, the percent strain imposed by an external species attached to the device at equilibrium can reach 92% upon fully elongating the device. The actual performance will, naturally, depend on the chemical details, but this serves as a reasonable estimate of the upper bound. Note also that the exact value is independent of the chosen length scale (5 Å in this case). Finally, the retractive force F may be tuned by cycling the electric field, as shown in Figure 6. A typical conformation was chosen from an ensemble of equilibrated hybrids in a high salt solution, (lBκ)-1 ) 0.42, and an electric field strength well above the critical value, EqlB/ ) -1. An immobile (dz/dt ) 0) phantom spring was attached to the chain end. The first plateau represents the time during which the field was on. The slight positive force exerted by the hybrid results because a configuration with X slightly less than Xe was chosen. When the field is turned off, the hybrid quickly retracts, exerting a force on the phantom spring. The field is then cycled on again, returning the retractive force to its original value. Thus, the actuator displays a lack of hysteresis as expected since no entanglement is observed between the linear and branched segments in this model. This is consistent with rheological studies of blends of hyperbranched and linear chains.18 The final plateau with a stronger retraction on the phantom spring results when the field polarity is reversed, effectively driving the dendritic and linear portions of the hybrid together. In conclusion, I have shown that a hybrid copolymer consisting of a dendritic and a linear block of equal but opposite charge operates as a tunable molecular actuator. The force required to unwind the linear block from the dendritic spindle and the equilibrium number of chain segments removed from the dendrimer are both tunable by the strength of an external electric field, moderated by the solution salt 1282
Figure 6. The retractive force F of the extended actuator as a function of time t. The hybrid was prepared in an extended state under a high field strength of EqlB/ ) -1 and high solution salt with (lBκ)-1 ) 0.42. Black symbols represent time periods during which the field was on and set to EqlB/ ) -1. Red indicates the force during which the field was set to zero. Blue corresponds to the time during which the polarity of the field was reversed such that EqlB/ ) +1. Typical snapshots for each condition are shown.
concentration. The device is free of hysteresis and appears to display a phase transition whose critical electric field strength also depends on the solution salt concentration. The construct is well within current synthetic capabilities and I hope that the results presented herein entice other researchers to produce a physical realization of this design. Acknowledgment. I thank Cynthia Welch, Kim Rasmussen, and Tommy Sewell for useful discussions and helpful comments during the preparation of this manuscript. This work is supported under the Los Alamos National Laboratory Directed Research and Development Program. LANL is operated by the University of California under Contract W-7405-ENG-36 for the U.S. Department of Energy. References (1) Davis, A. P. Nature 1999, 401, 120. Madden, J. D. W.; Madden, P. G. A.; Hunter, I. W. Proceedings of SPIE on Smart Structures and Materials: ElectroactiVe Polymer Actuators and DeVices (EAPAD) 2002, 4695, 176. Smela, E. AdV. Mater. 2003, 15, 481. Mavroidis, C.; Dubey, A.; Yarmush, M. L. Annu. ReV. Biomed. Eng. 2004, 6, 363.
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(2) Cooke, R. Physiol. ReV. 1997, 77, 671. (3) Kukowska-Latallo, J. F.; Bielinska, A. U.; Johnson, J.; Spindler, R.; Tomalia, D. A.; Baker, J. R. Proc. Natl. Acad. Sci. U.S.A 1996, 93, 4897. Ottaviani, M. F.; Sacchi, B.; Turro, N. J.; Chen, W.; Jockush, S.; Tomalia, D. A. Macromolecules 1999, 32, 2275. Li, Y.; Dubin, P. L.; Spindler, R.; Tomalia, D. A. Macromolecules 1995, 28, 8426. Kabanov, V. A.; Zezin, A. B.; Rogacheva, V. B.; Gulyaeva, Zh. G.; Zansochva, M. F.; Joosten, J. G. H.; Brackman, J. Macromolecules 1999, 32, 1904. (4) Welch, P.; Muthukumar, M. Macromolecules 2000, 33, 6159. (5) Iyer, J.; Fleming, K.; Hammond, P. T. Macromolecules 1998, 31, 8757. Stewart, G. M.; Fox, M. A. Chem. Mater. 1998, 10, 860. Emrick, T.; Hayes, W.; Fre´chet, J. M. J. Polym. Sci. A: Polym. Chem. 1999, 37, 3748. Roovers, J.; Comanita, B. AdV. Polym. Sci., 1999, 142, 179. Wang, H.; Simon, G. P.; Hawker, C.; Tiu, C. Mater. Res. InnoVat. 2002, 6, 160. (6) Gitsov, I.; Fre´chet, J. M Macromolecules 1993, 26, 6536. (7) Jeong, M.; Mackay, M. E.; Vestberg, R.; Hawker, C. J. Macromolecules 2001, 34, 4927. (8) Mackay, M. E. C. R. Chimie 2003, 6, 747. (9) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987.
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(10) The choice of the Bjerrum length as the length scale for the simulations is natural because the length between branch points in dendrimers is of the order 5 Å and the equilibrium bond length chosen between the branches in the simulations is 0.7lB ≈ 5 Å. (11) Muthukumar, M.; Welch, P. Polymer 2000, 41, 8833. (12) Jarzynski, C. Phys. ReV. Lett. 1997, 78, 2690. Jarzynski, C. Phys. ReV. E 1997, 56, 5018. (13) Hummer, G.; Szabo, A. Proc. Nat. Acad. Sci. U.S.A. 2001, 98, 3658. (14) Gore, J., Ritort, F., Bustamante, C. Proc. Nat. Acad. Sci. U.S.A. 2003, 100, 12546. (15) The minima fall around roughly 2kBT because the ground state was defined as having the chain end precisely aligned along the axis of extension, but lateral excursions are included in the statistics and thus raise the average magnitude of the minima. (16) Grosberg, A. Y.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994. (17) Chaˆtellier, X.; Senden, T. J.; Joanny, J.-F.; di Meglio, J.-M. Europhys. Lett. 1998, 41, 303. (18) Nunez, C. M.; Chiou, B.-S.; Andrady, A. L.; Khan, S. A. Macromolecules 2000, 33, 1720.
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