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Piston-Rotaxanes as Molecular Shock Absorbers E. M. Sevick*,† and D. R. M. Williams*,‡ †
Research School of Chemistry and ‡Research School of Physical Science & Engineering, Australian National University, Canberra 0200 ACT, Australia Received October 8, 2009. Revised Manuscript Received January 20, 2010
We describe the thermomechanical response of a new molecular system that behaves as a shock absorber. The system consists of a rodlike rotaxane connected to a piston and tethered to a surface. The response of this system is dominated by the translational entropy of the rotaxane rings and can be calculated exactly. The force laws are contrasted with those for a rigid rod and a polymer. In some cases, the rotaxanes undergo a sudden transition to a tilted state when compressed. These pistonrotaxanes provide a potential motif for the design of a new class of materials with a novel thermomechanical response.
Introduction Rotaxanes are molecules that are architecturally similar to some baby rattles: one or more ringlike molecules threaded onto a molecular axle that is capped on both ends with large stoppers to prevent the rings from sliding off.1,2 These molecules were first synthesized over 40 years ago, and they are now routinely synthesized in large quantities. Rotaxanes also occur naturally because the structure has been recognized in some proteins. They can be small molecules or, using longer axles, larger polyrotaxanes, and there is an endless zoo of possible architectures. Rotaxane molecules also lend themselves to exact analysis by equilibrium statistical mechanics. Previous descriptions have focused predominantly upon “slip-link” behavior,3-8 where the axle slips through a single threaded ring or link. However, here we focus on a rotaxane molecule that is intrinsically different from these slip-link systems. In our system, there are numerous free rings whose translational entropy dominates the rotaxanes’ behavior. Here we show one example of how the response of the confined rings can be used to design molecules with particular thermomechanical properties. In particular, we couple an external piston to one of the rings of a rotaxane molecule (Figure 1) and exploit the gaslike entropy of the other rings to determine the piston’s elastic response. We refer to this molecular system, where the piston can compress the axle-bound rings, as a piston-rotaxane. In this article, we describe a surface-tethered piston-rotaxane that behaves as a molecular version of an automobile shock absorber. The piston-rotaxane system consists of a cylindrical axle of length C (Figure 1). Threaded onto the axle are n þ 1 rings, each taking up a length d along the axle and free to move. The (n þ 1)th or final ring is the piston ring, located at distance X along the axle and attached to a piston rod of length P. The total extension of the grafted piston-rotaxane from end to end is thus L=X þ P. A collection of such piston-rotaxanes, each tethered by one end to a surface, would give a spongy surface layer that resists compression due to
the entropy of the rings. Indeed, this is somewhat similar to surfacegrafted polymer layers, although there are significant differences. In the case of grafted polymer layers, the conformational entropy of the chains prevents compression. In the case of piston-rotaxanes, it is the translational entropy of the rings on the axle that has the same effect. However, as we will show, because the rings are free to slide, whereas in a polymer each monomer is attached to its neighbor, a piston-rotaxane is a much more efficient spring than a polymer (i.e., the effective elastic force per ring is much greater than the force per monomer in a polymer).
Force Laws for a Single Piston-Rotaxane Molecule The partition function, ZX, for the gas of rings with the piston ring fixed at X is given by n Z x2 -d Z X -d 1 d X - -nd dx1 ::: dxn ¼ ð1Þ ZX ¼ n! 2 d=2 xn -1 þ d where xi is the position of the ith ring and x(n þ 1) = X. This leads to a free energy of F = -kBT ln ZX and hence to an average force on the piston ring due to the other n rings of Æ f æ = kBT(∂ ln ZX/∂X) or 0 1 -1 X d A ð2Þ Æ f æ ¼ kB Tn@ 2 - nd This is the mean force on the piston ring when it is at a fixed position X along the axle and is identical to the mean “pressure” of a 1D ideal gas of n impenetrable molecules corrected for the fact that the free length is not X but X - d(n þ 1/2). However, in this single-molecule system, the number of rings, n, may well be small. To see what effect this has we ask, what is the mean displacement, ÆXæ, for an applied constant force, f ? The partition function for this case is Z Zf ¼
*E-mail:
[email protected];
[email protected]. (1) Huang, F.; Gibson, H. Prog. Polym. Sci. 2005, 30, 982–1018. (2) Takata, T.; Kihara, N.; Furusho, Y. Polyrotaxanes and Polycatenanes: Recent Advances in Syntheses and Applications of Polymers Comprising of Interlocked Structures. In Polymer Synthesis; Advances in Polymer Science Series; Springer: Berlin, 2004; Vol. 171, pp 1-75. (3) Mansfield, M. Macromolecules 1991, 24, 3395–3399. (4) Rieger, J. Mcromoelcules 1989, 22, 4540–4544. (5) Loomans, D.; Sokolov, I.; Blumen, A. Macromolecules 1996, 29, 4777–4781. (6) Baulin, V.; Johner, A.; Marques, C. Macromolecules 2005, 38, 1434–1441. (7) Baulin, V.; Lee, N.; Johner, A.; Marques, C. Macromolecules 2006, 39, 871–876. (8) Sommer, J. J. Chem. Phys. 1992, 97, 5777–5781.
5864 DOI: 10.1021/la903801x
C -d=2 ðn þ 1=2Þd
dX ZX ðXÞ expð -βfXÞ
ð3Þ
with β 1/(kBT). To evaluate this, we write u = X - (n þ 1/2)d and obtain Zf ¼ ðβf Þ -ðnþ1Þ e -βf ðnþ1=2Þd ½1 -e -δ en ðδÞ
ð4Þ
with δ βf(C-(n P þ 1)d). Here, en(x) is the exponential sum function or en(x) nk=0 (xk/k!). The probability distribution for
Published on Web 02/16/2010
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Figure 2. Scaled piston-ring position, X/C, versus the dimensionless force, fC/kBT, for a piston-rotaxane with n = 10 and 1 free rings with a ring thickness of d = 1/20C. The solid lines provide the force law under fixed or applied force (i.e., ÆXæ/C versus applied fC/ kBT), and the dashed lines are for a fixed piston position, or X/C versus ÆfæC/kBT.
Equation 7 also permits a simple calculation of the response of the piston-rotaxane to a weak applied force. The spring constant, k, for the deviation of the mean position of the ring piston from ÆXæf=0, defined as f = -k(ÆXæ - Xf=0), is given by Figure 1. Schematic of a molecular shock absorber constructed from a piston attached to a ring of a rotaxane molecule that is surface-grafted. The extent of the molecular shock absorber, L = X þ P, fluctuates, imparting springlike thermomechanical properties to a surface with grafted piston-rotaxanes.
the displacement, X, under an applied force, f, is Pf(X) = ZX exp(-βfX)/Zf, where the mean displacement is found from R C -d=2 ∂ ln Z ÆXæ ¼ ðn þ 1=2Þd dX XPnf ðXÞ ¼ -β -1 ∂f f . Using the identities den(x)/dx = en - 1(x) and en(x) = en - 1(x) þ xn/n!, we find ÆXæ ¼ ðn þ 1Þðβf Þ
-1
1 þ n þ d þ Xc 2
ð5Þ
where
1 Xc ¼ - δnþ1 ðβf Þ -1 ðeδ -en ðδÞÞ -1 ð6Þ n! The mean displacement of the piston ring, ÆXæ, at an applied force, f, is very different from the fixed displacement, X, at a mean force, Æ f æ, as shown in Figure 2. In a macroscopic system, where the number of rings is very large, there would be an infinitesimal difference in these two curves, but for a microscopic system where the number of rings is small enough that the fluctuations of each ring contributes significantly to the fluctuations in the system,9 the difference is appreciable. To examine this further, consider the case of infinitely thin rings, d = 0, where the force applied to the piston ring is weak or βfC , 1 and eqs 5 and 6 resolve to nþ1 βfC 1ð7Þ ÆXæd ¼0 ¼ C nþ2 ðn þ 2Þðn þ 3Þ For zero applied force, this predicts the average equilibrium position of the piston ring to be ÆXæf=0 = C[(n þ 1)/(n þ 2)]. For a macroscopic system with n . 1, this piston ring would be located at C or the end of the axle. For microscopic systems, say n = 1, the equilibrium position of the piston ring would be ÆXæf=0 = 2C/3 (i.e., the piston ring and the only other ring in the system divide the axle equally between them). Of course, in the case of n = 0 when the piston ring is the sole ring, ÆXæf=0 = C/2. (9) Wang, G.; Sevick, E.; Mittag, E.; Searles, D.; Evans, D. Phys. Rev. Lett. 2002, 89, 050601–050604.
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k ¼ kB TC -2
ðnþ2Þ2 ðn þ 3Þ nþ1
ð8Þ
Thus even in the case of no added rings or n = 0, there is an elastic response of the piston ring with k = 12kBTC-2 caused by the entropy of confinement of the ring to the finitely sized axle. For large n, the spring constant is k = kBTC-2n2. This n2 dependence can be understood as follows: because the spring constant arises solely from entropic confinement, it must be proportional to kBT and must be divided by the square of the length that characterizes the confinement. For large n, the confinement length is C/n because each ring is confined to an equal portion of the axle. The spring constant can be used to find the approximate size of the equilibrium fluctuations at the position of the piston ring, X. Using equipartition, 1 1 kÆðX -ÆXæÞ2 æ ¼ kB T ð9Þ 2 2 the size of the fluctuations under zero applied force is characterized by rffiffiffiffiffiffiffiffiffiffiffi C nþ1 2 ð10Þ ÆðX - ÆXæf ¼0 Þ æ ¼ nþ2 nþ3 Thus, for n = 0 the piston ring fluctuates freely over the axle length C, whereas for a large number of rings the piston ring travels only as far as the nearest free ring, or on the order of C/n. In the opposite limit of large forces βfC . 1, or more accurately βfC . n ln(βfC), the “correction” term Xc is small and we are left with ÆXæf = (n þ 1)kBTf-1. In this case, the gas is strongly compressed at the bottom end of the axle and the length C is irrelevant. Of course, the size of the fluctuations for any applied force and ring thickness d can be calculated from the exact partition function and the equation ÆX 2 æ -ÆXæ2 ¼ β -2
∂2 lnðZf Þ ∂f 2
ð11Þ
Surface-Grafted Piston-Rotaxane We now consider the end tethering of the piston-rotaxane to a flat surface and assume that there are no interactions among the grafted piston-rotaxane molecules, which, for rotaxanes that are DOI: 10.1021/la903801x
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freely rotating at the grafting point, implies that the grafting density is less than P-2. The piston-rotaxanes are compressed by another flat, parallel surface that is at a distance H the compression distance, above the grafting surface. This upper plate is not attached to the rotaxanes, and the surfaces form a slit of thickness, H, that confines the molecule. The response of the molecule depends critically upon the anchoring conditions at the grafting surface and, in particular, the angle between the axle and the surface normal, denoted θ. There are three cases to consider: fixed or rigid anchoring orthogonal to the grafting plane, θ = 0; freely rotating grafting where θ and the associated azimuthal angle φ are free to rotate in the absence of interference by the compressing plane; and grafting where orientation/rotation is possible and associated with a potential penalty U(θ). The first case, where the piston-rotaxane is grafted rigidly and orthogonally, is the simplest and allows us to introduce the important length scales for the other cases: the maximum length of the molecular shock absorber Λ P þ C - d/2 and the minimum length λ P þ (n þ 1/2)d. The force law for the rigidly, orthogonally grafted piston rotaxane follows readily from eq 2 (i.e., f = 0 for compression distance H > Λ and Æ f æ = kBTn(H - λ)-1 for H < Λ.
piston-rotaxane, the partition function is independent of the height and so f(H > Λ) = 0. In the strong compression regime (3), where the compression distance is smaller than the minimum length of the molecule, the partition function reduces to Z3(H) = Hg, where g is independent of H and the force is exactly what one obtains for a rigid rod (i.e., f(H > λ) = kBT/H) and is completely independent of the molecular details of the shock adsorber. However, under moderate compression, whenever the compression distance is between λ and Λ, the compression force depends upon the molecular characterstics of the piston-rotaxane. The partition function in this regime is simplified by reversing the order of integration in the first term and integrating over μ, giving Z Z2 ðHÞ ¼
Z Z3 ðHÞ ¼
Z
H=λ
H=μ
dμ H=Λ
λ
0
λ
Z dL ZL þ
Z
H=Λ
Λ
dμ 0
λ
dL ZL ð13Þ
The equilibrium compressive force is f = kBT∂ ln Z/∂ H. When the compression distance is larger than the maximum extent of the (10) Miller, I.; Williams, D. Phys. Rev. E 2000, 61, R4706–R4709.
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Λ
dL L -1 ZL
ð14Þ
H
RH dL ZL ðLÞ kB T ¼ H þRΛλ f2 ðHÞ dL L -1 ZL ðLÞ
ð15Þ
H
Before considering the piston-rotaxane to be freely rotating at the grafting point, it is first instructive to examine an inextensible or rigid rod of fixed length L0, similarly grafted.10 The difference in these two systems is that the piston rotaxane’s length varies over λ e L e Λ whereas the rigid rod is of fixed length L = L0. If the slit thickness is larger than the rod length or H > L0, then the rigid rod can rotate freely around its grafting point and the partition function by the solid angle that the rod can occupy R is Rgiven 1 Z = 2π 0 dφ 0 dμ = 2π, where μ cos(θ). If the slit thickness is smaller than the rod length or H > L0, then volume exclusion restricts the range of angles to 0 < μ < H/L0 and the partition function is then Z = 2πH/L0. The force on the compressing plate, f = kBT∂ ln Z/∂H, is 0 if H > L0 and kBT/H if H < L0. Note in particular that for H < L0 the force is independent of the length of the rigid rod, L0, and depends only upon H. For rotaxane, the rod length is not fixed but varies with the position of the piston ring, and this alters the simple rigid rod results. Let ZL be the partition function for the piston-rotaxane with length L (i.e., by eq 1, ZL = [1/n!](L - λ)n). There are three compression regimes depending upon how the compression distance compares with the length scales of the problem: (1) no compression, when the slit thickness is larger than the maximum possible extent of the molecule or H > Λ; (2) moderate compression, when the slit thickness is between the minimum and maximum possible molecular extents or λ < H < Λ; and (3) strong compression, when the slit thickness is less than the minimum extent or H < λ. In the nocompression regime (1), the partition function is independent of compression distance, H. In the other two cases, moderate and strong compression, the partition functions are Z 1 Z H=μ Z H=Λ Z Λ dμ dL ZL þ dμ dL ZL ð12Þ Z2 ðHÞ ¼ λ
λ
Z dL ZL þ H
which then yields a particularly simple form for the reciprocal of the force,
Freely Rotating End-Grafted Rotaxane
H=Λ
H
This force recovers the 1/H term expected for a rigid rod with an additional term arising from the internal structure of the pistonrotaxane. Two things are evident from the equation and its predictions in Figure 3. First, under strong compression, or as H f λ, the force reduces to the rigid rod result, f = kBT/H, so that there is no discontinuity in the compressive force. Second, under moderate compression, or as H f Λ, the force falls smoothly to zero so that unlike the rigid rod case there is no!discontinuity for weak comn P n k -1 ð -λÞn -k Lk allows us to pression. Defining gn ðLÞ ¼ k ¼1 k write the reciprocal of the force, kBT/f2, as Hþ
ðH -λÞnþ1 ½ð -λÞn lnðΛ=HÞþgn ðΛÞ -gn ðHÞ -1 nþ1
ð16Þ
From eq 15, we find that near H = Λ f2 ¼ ðn þ 1ÞkB TðΛ -HÞΛ -1 ðΛ -λÞ -1
ð17Þ
so that the spring constant for small compressions is k = (n þ 1)kBTΛ-1(Λ - λ)-1. This also tells us at what compression the force begins to cross over to the rigid-rod behavior of f = kBT/H: at a compression distance of roughly (Λ - λ)/(n þ 1) (i.e., the distance the piston ring has to move to hit the nearest free ring). Thus, for large n the system behaves as a rigid rod except at the weakest compressions.
Grafted with a Tilt/Rotational Potential Now we consider the addition of a potential that depends on the angle θ between piston-rotaxane and the surface normal. We consider the simplest potential, U = -RkBTμ (R > 0 and μ cos(θ)), which gives an energy contribution proportional to θ2 at small tilt angles. Again, we first consider a rigid rod of fixed length L0 with this rotational penalty. The partition function is Z
H=L0
ZðHÞ ¼
dμ eRμ ¼ R -1 ½eRH=L0 -1
ð18Þ
0 -RH/L0 -1 which gives a force for H < L0 of f = kBTRL-1 ] . 0 [1 - e For small R this is f ≈ kBTH-1(1 þ (RH/2L0)) whereas for large R -RH/L0 ]. Note that for large R the we have f ≈ kBTRL-1 0 [1 þ e
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Figure 3. Compression force, f/kBT, versus the compression distance, H, for a surface-grafted piston-rotaxane with n = 0 free rings (-b-), n = 1 (---), n = 5 (-4-), n = 10 (---), and n = 50 ( 3 3 3 ) and for a surface-grafted rigid rod of length Λ (;) where f/kBT = 1/H. The piston-rotaxanes and rigid rod are freely rotating at the grafting point. To contrast the force profiles of rotaxanes with the different number of rings, n, it is convenient to consider rotaxanes whose minimum and maximum extents, λ and Λ, are fixed irrespective of the number of rotaxane rings. Here we have chosen λ = 2 and Λ = 6 with varying n. Once these are known P, C, and d can be determined, but in general they will be different for the different curves.
force becomes independent of the compression distance H and depends on the rod length, in direct contrast to a freely rotating rigid rod. The discontinuous force or “jump force” at contact is f ðL0 Þ ¼ KTRL0-1 ½1 -e -R -1
ð19Þ
which is larger than the compressive force at the same H for a freely rotating rigid rod. For small R this jump force is f(L0) ≈ kTL0-1(1 þ R/2) whereas for large R we obtain kTL-1 0 R. When the rigid rod is replaced by the piston-rotaxane so that the length is no longer fixed, an exact calculation of the partition function is possible along the same lines as the zero potential case presented above. However, for brevity, we focus here on a novel effect of the potential that occurs when both R and n are large: a discontinuous transition between a tilted, fully extended pistonrotaxane and a nontilted (orthogonal), shortened piston-rotaxane. Here we neglect fluctuations and write down the free energy directly as the sum of a torque term and a compression term. It is then F H -λ ¼ -Rμ -n ln kB T μ
ð20Þ
For R . 1, the weakly compressed piston-rotaxane remains vertical at μ = 1. However, at some critical compression H (1 λ/H) = g1, and the second condition, where the mimimum remains but is now local rather than global, gives n/R > (1 H/λ)/ln((Λ - λ)/(H - λ)) = g2. Figure 4 is a phase diagram based on this simple free-energy model, cast as n/R, representing the ratio of entropic to potential-energy parameters, versus H/Λ Langmuir 2010, 26(8), 5864–5868
Figure 4. Phase diagram of a compressed piston-rotaxane molecule, grafted with an angular potential. n/R characterizes the entropy of the free rings relative to the strength of the rotational potential. Under weak compression (H/Λ ≈ 1), the tilt energy dominates the ring entropy, and for all cases except those where the number of free rings is very large, the molecule will not tilt but its piston ring will compress the free rings with the applied compressive force. However, at a critical compression distance (;), the piston ring will release the compressed rings, the molecular extension will increase, and the molecule will tilt.
representing the compression distance scaled by the largest molecular length scale of the problem. For H/Λ > 1, the pistonrotaxane is uncompressed and untilted. For H/Λ
