J. Phys. Chem. B 1998, 102, 5699-5703
5699
Physical Basis of Self-Assembly Macrocyclizations Gianfranco Ercolani* Dipartimento di Scienze e Tecnologie Chimiche, UniVersita` di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy ReceiVed: March 30, 1998
A theoretical treatment of self-assembly macrocyclizations occurring under thermodynamic control is presented. The fundamental quantities on which the treatment is based are the effective molarity of the self-assembling cyclic n-mer (EMn) and the equilibrium constant for the intermolecular model reaction between monofunctional reactants (Kinter). Knowledge of these quantities allows the evaluation of the distribution curve of the selfassembling macrocycle. In order for effective self-assembly to take place two conditions are required: (i) the self-assembling macrocycle must have an EM much larger than that of the other cyclic oligomers; (ii) the product EMnKinter must be not lower than 185r, where r is the number of bonds that hold together the monomer units in the cyclic oligomer, the higher the better. It is shown that in the limit of an infinite value of Kinter there is a critical monomer concentration (cmc ) nEMn) below which the system is virtually composed of the self-assembling macrocycle only and above which the concentration of the latter remains constant and the excess monomer produces acyclic species only. In general the optimum monomer concentration for self-assembly is slightly more than one-tenth of the cmc. Deviation from this value is less and less important the higher the value of EMnKinter; however, the concentration of the initial monomer should not be outside the range defined by the lower self-assembly concentration and the cmc. Previous conclusions about self-assembly macrocyclizations drawn by Hunter et al. (J. Chem. Soc., Chem. Commun. 1995, 2563) are criticized in the light of the present approach.
Introduction Self-assembly consists of the spontaneous generation of a well-defined, discrete supramolecular architecture from a given set of components under thermodynamic equilibration.1 Application of this phenomenon to macrocyclization is quite attractive since cyclization of a large molecule by the usual synthetic methods, employing kinetically controlled reactions, is frequently a tedious and low-yield process. Self-assembly macrocyclizations are schematically represented, by way of illustration, in parts a and b of Figure 1 which respectively show the self-assembly of a bifunctional monomer into a cyclic tetramer and of two symmetrical monomers into a cyclic dimer. Despite the numerous examples of self-assembling systems that can be found in the literature, the physicochemical basis of selfassembly has not received, apart from some exceptions,1b,2 adequate attention. Some years ago we presented a general treatment of the problem of macrocyclization under thermodynamic control that was applied to the cyclization of flexible chains.3 Since selfassembly macrocyclization is just a particular case of the general problem of macrocyclization under thermodynamic control, we wish to report on the application of this treatment to the problem of self-assembly with the aim of presenting a unified view of the subject as well as showing results that should prove useful both to an understanding of the process and to the design of new self-assembling systems. Treatment of Self-Assembly Macrocyclizations Self-Assembly of a Bifunctional Monomer. To apply our treatment to the problem of self-assembly, it is useful to briefly * Fax: +39 6 72594328. E-mail:
[email protected].
Figure 1. Self-assembly by predisposed building blocks: (a) selfassembly of a bifunctional monomer into a cyclic tetramer; (b) selfassembly of two symmetrical monomers into a cyclic dimer.
recall its principal features.3 Consider a bifunctional monomer A-B (M1) bearing two different functional groups, each capable of reacting with the other only in a reversible addition reaction. After equilibration, a system initially composed of monomer units M1, contains in principle an infinite number of cyclic oligomers Ci as well as an infinite number of linear oligomers Mi, i being the polymerization degree. This can be expressed by the mass balance equation ∞
[M1]0 )
∑ i)1
∞
i[Ci] +
i[Mi] ∑ i)1
(1)
where [M1]0 is the initial monomer concentration. In our previous paper it has been shown that eqs 2 and 3 hold
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5700 J. Phys. Chem. B, Vol. 102, No. 29, 1998
Ercolani
xi Kinter
(2)
[Ci] ) EMixi
(3)
[Mi] )
where Kinter is the equilibrium constant for the intermolecular model reaction between the functional groups A and B which is assumed to be identical for all the association equilibria between the linear oligomers, EMi is the equilibrium effective molarity4 defined as K(intra)i/Kinter where K(intra)i refers to the equilibrium of cyclization of the acyclic i-mer Mi to yield the cyclic i-mer Ci, and x is the fraction of reacted end groups in the acyclic part of the polymer; for our purposes, however, x is just a function of [M1]0, confined to the range 0 e x < 1. Introducing eqs 2 and 3 into eq 1, eq 4 is obtained ∞
[M1]0 )
∑ i)1
iEMixi +
∞
1
∑ixi
Kinter i)1
(4)
Since the second summation appearing in the right-hand side of eq 4 is equal to x/(1 - x)2, eq 4 is transformed into eq 5. ∞
[M1]0 )
iEMixi + ∑ i)1
x Kinter(1 - x)2
(5) SCHEME 1
Eqs 2, 3, and 5 are the master equations to be used in any description of cyclizations under thermodynamic control involving the addition of a monomer of the type A-B. In the case of flexible chains following Gaussian statistics and leading to strainless rings, the Jacobson-Stockmayer theory predicts that EMi values are proportional to i-5/2.5 The case of equilibrium macrocyclization of flexible chains with allowance for strain in the first cyclic oligomers has been discussed in our previous work.3 Under these conditions a wide distributions of cyclic oligomers is generally obtained unless the reaction is carried out in very dilute solution promoting the formation of the cyclic monomer only. By contrast self-assembly takes place when the monomer has a rigid structure predisposed6 in such a way that formation of all the cyclic oligomers except one is prevented by strain or, in the case of high cyclic oligomers, by unfavorable entropic factors. This is equivalent to saying that the EM of a certain cyclic oligomer Cn is so large with respect to those of the others that ∞
iEMixi ≈ nEMnxn ∑ i)1
(6)
This condition, which expresses the stability of one macrocycle over all the others is the first condition to be met for self-assembly, namely, the condition of selectivity. Thus eq 5 reduces to eq 7.
[M1]0 x ) nxn + EMn EMnKinter(1 - x)2
Figure 2. Plots of the yield of a self-assembled cyclic tetramer, as that sketched in Figure 1a, against [M1]0/EM4. The three curves from the bottom up, refer to EM4Kinter ) 10, 100, and 1000, respectively. The limiting curve in bold refers to the case in which Kinter tends to infinity.
(7)
Eq 7 with n g 2 (the case n ) 1, referring to the formation of the cyclic monomer, is outside the definition of self-assembly) together with eqs 2 and 3 (with i ) n) refers to the case in which an unlimited number of linear oligomers and only one cyclic oligomer are in conditions of thermodynamic equilibration (Scheme 1). This scheme differs from that previously proposed by Hunter et al.2 because in the latter only a limited number of acyclic oligomers (up to n + 2; no equations given) was
considered for the evaluation of oligomer distributions. Such a simplified scheme can be in serious error in the region of high x values. The yield of the self-assembling macrocycle, given by 100n[Cn]/[M1]0, is expressed, taking into account eqs 3 and 7, by eq 8.
(
Cn% ) 100nxn/ nxn +
)
x EMnKinter(1 - x)2
(8)
Of course the complement to 100 of Cn% represents the yield of the acyclic fraction. Equations 7 and 8 can be used to build plots of Cn% vs [M1]0/EMn which only depend on EMnKinter. This is done by compiling a list of x values in the range 0 e x < 1 and then calculating, for a given value of the product EMnKinter, the couple of [M1]0/EMn and Cn% values corresponding to each x, by eqs 7 and 8, respectively. Such plots are shown, by way of illustration, in Figure 2 for the cases n ) 4 and EMnKinter ) 10, 100, and 1000, respectively. Plots obtained for other n values are qualitatively similar. It is apparent from Figure 2 that the driving force for self-assembly is given by the product EMnKinter; a greater value of EMnKinter is beneficial not only to the yield of the self-assembling macrocycle but also to the amplitude of the concentration range over which selfassembly takes place. It is also evident from the abscissa of the graph in Figure 2 that the actual range directly depends, for a given value of EMnKinter, on the value of EMn. It is interesting to consider the limiting case in which Kinter approaches infinity. Under this condition the second term in the right-hand side of eq 7, representing the acyclic fraction, is negligible until x ≈ 1, this means that Cn% is always equal to 100% until [Cn] reaches EMn, its maximum possible value (see eq 3); this occurs when [M1]0/EMn ) n (see eq 7). The acyclic fraction begins to grow for values of [M1]0/EMn exceeding the
Physical Basis of Self-Assembly Macrocyclizations
J. Phys. Chem. B, Vol. 102, No. 29, 1998 5701
value of n; when this condition is met Cn% ) 100nEMn/[M1]0. The limiting profile of Cn% when Kinter tends to infinity, also reported in Figure 2, shows a cusp corresponding to [M1]0 ) nEMn; this is the so-called critical monomer concentration (cmc), analogous to that discussed in the macrocyclization of flexible chains,3,5 below which the system is composed of only the cycle Cn, and above which the concentration of the cycle remains constant and the excess monomer produces acyclic species only. Thus from a synthetic point of view there is no advantage to carry out the preparation of a self-assembling macrocycle above the cmc. Hunter et al. pointed out that the upper limit of the concentration range over which a self-assembled structure is stable is given by EMn.2 Our analysis shows, in contrast, that the upper limit is given by the cmc, i.e., by the product nEMn. Eq 8 has a maximum in correspondence of
x)
n-1 n+1
(9)
By substitution of eq 9 into eq 8, eq 10 is obtained
(
)
an (Cn%)max ) 100/ 1 + EMnKinter
(10)
where a ) (n + In order for selfassembly to be virtually complete, it is required that (Cn%)max > 99%; this occurs when EMnKinter g 100an. Since a is confined in the very narrow range 1.69 e a < e2/4 ()1.85) for 2 e n < ∞, this condition can be translated in the useful form 1)n+1/[4n2(n
1)n-1].
EMnKinter g 185n
(11)
that can be considered as the second condition for self-assembly, namely, the condition of stability of the self-assembling macrocycle over the acyclic oligomers. This is an important relation showing that the required driving force for self-assembly directly depends on the number of monomer units constituting the cyclic oligomer. The plots reported in Figure 2 illustrate this point; only the plot relative to EMnKinter ) 1000 satisfies the condition in eq 11. The value of [M1]0/EMn corresponding to (Cn%)max is easily found (eq 12) by introducing eq 9 into eq 7 and taking into account the definition of a.
( ) ( [M1]0 EMn
)n 1+
max
)( )
an n-1 EMnKinter n + 1
n
(12)
When the condition of self-assembly (eq 11) holds, the value between the first parentheses in the right-hand side of eq 12 is ≈1. Moreover, since the term [(n - 1)/(n + 1)]n lies in the very narrow range from 0.111 to e-2 ()0.135) for 2 e n < ∞, it can be concluded that the optimum monomer concentration for self-assembly is slightly more than one-tenth of nEMn, the cmc. Hunter et al. introduced the concept of critical self-assembly concentration (csac) as the concentration at which the complex is half-assembled, i.e., when its yield is 50%.2 It is evident by inspection of Figure 2 that there are two monomer concentrations at which the complex is half-assembled. Although not explicitly expressed by Hunter and co-workers, they refer to the csac as the lower one. It is also evident that this concentration does not refer to a critical phenomenon characterized by a sharp change in the behavior of the system, as that observed at the cmc. Thus for both these reasons we believe that the term lower self-assembly concentration (lsac) is more
appropriate than csac. Reserving the adjective critical to the cmc, it also avoids any possible source of confusion between the two concepts. Apart from semantics, it is evident that the concept of lsac makes sense only under self-assembly conditions, namely when eq 11 holds. The lsac coincides with [M1]0 when the cyclic and acyclic fractions contain the same number of monomer units, i.e., when
nEMnxn )
x Kinter(1 - x)2
(13)
Although this equation cannot be solved analytically, it is always possible to obtain a numerical solution for x by the NewtonRaphson method.7 However, it is useful to consider the case in which x/(1 - x)2 ≈ x. This approximation, which is equivalent to considering that the acyclic fraction is composed of the monomer only (see eq 2), can be considered sufficiently accurate for practical purposes when x e 0.1. Introducing this condition into eq 13, one can see that the approximation x/(1 - x)2 ≈ x is verified when
EMnKinter g (0.12/n)10n
(14)
This condition is satisfied when eq 11 holds only for n e 4 but becomes increasingly more demanding than the condition in eq 11 on going up from n > 4. When the condition in eq 14 holds, eq 13 can be simplified and solved analytically to yield
x ) 1/(nEMnKinter)1/(n-1)
(15)
Introducing eq 15 into eq 7, one obtains
lsac )
2 n1/(n-1)EM1/(n-1) Kn/(n-1) n inter
(16)
This equation is equal to that obtained by Hunter et al. apart from the presence of the factor 2 in the numerator.2 This is due to the fact that the authors in their derivation erroneously considered the lsac equal to the actual concentration of the monomer in solution; of course, to be of any utility, the lsac must be expressed in terms of the initial monomer concentration. Apart from this minor flaw, the authors made the observation, based on the equation analogous to eq 16, that when n tends to infinity the lsac tends toward a limiting value of 1/Kinter. This led them to the paradoxical conclusion that “Such large assemblies will have apparent stability constants which are the same as the anticipated stability constant for the formation of open polymeric assemblies (Kinter), but they will be stable with respect to polymerization over a reasonable concentration range”.2 In fact the authors failed to appreciate that eq 16 is an approximate equation which is valid only under the constraint of eq 14. In order for eq 16 to maintain its validity when n tends to infinity, Kinter must also tend to infinity. Thus the conclusion that for large assemblies the limiting value of the lsac is 1/Kinter is without foundation. In fact eq 16 provides a value of the lsac which can be badly underestimated with respect to the actual one if the condition in eq 14 is not satisfied. Self-Assembly of Two Symmetrical Monomers. Selfassembly of a large macrocycle by cyclooligomerization of two different symmetrical momomers is another typical case which deserves consideration. Let us consider the equilibration of an equimolar mixture of two of such monomers, A-A and B-B. After equilibration three type of acyclic oligomers will be present in solution, namely, A-(AB-BA-)i-1A ()MiA), B-(BA-AB-)i-1B ()MiB), and A-A(B-BA-A)i-1B-B
5702 J. Phys. Chem. B, Vol. 102, No. 29, 1998
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()Mi), together with the cyclic oligomers Ci formed by cyclization of Mi. In our previous paper it has been shown that, taking the fraction of reacted end groups in the acyclic part of the polymer equal to x1/2, the distribution of Mi and Ci is still given by eqs 2 and 3, respectively, whereas the distribution of MiA ()MiB) is given by eq 17.3
[MiA] )
i-0.5
x 2Kinter
(17)
Thus the mass balance equation takes the form of eq 18.3 ∞
[M1A]0 ) [M1B]0 )
iEMixi + ∑ i)1
1/2
x
2Kinter(1 - x1/2)2
(18)
When the first condition for self-assembly (eq 6) is taken into account eq 18 reduces to eq 19.
[M1A]0 x1/2 ) nxn + EMn 2EMnKinter(1 - x1/2)2
(19)
The yield of the self-assembling macrocycle, given by 100n[Cn]/[M1A]0, is expressed by eq 20.
(
Cn% ) 100nxn/ nxn +
)
x1/2 2EMnKinter(1 - x1/2)2
(20)
Plots of Cn% vs [M1A]0/EMn evaluated by eqs 19 and 20 are qualitatively similar to the plots shown in Figure 2 for the A-B reaction. Eq 20 has a maximum in correspondence of
x)
(2n2n +- 11)
2
(21)
Substituting eq 21 into eq 20 an equation of the same form as eq 10 is obtained with a ) (2n + 1)2n+1/[8n2(2n - 1)2n-1]. Since a is confined in the very narrow range 3.37e a < e2/2 ()3.69) for 1 e n < ∞ (here the case n ) 1 is included in the definition of self-assembly), the second condition for self-assembly becomes
EMnKinter g 369n
(22)
It is interesting to note that a cyclic n-mer, if formed by oligomerization of A-B has n AB bonds, whereas if formed by oligomerization of A-A and B-B has 2n AB bonds. Thus the second condition for self-assembly, as expressed by eqs 11 and 22, can be summed up by eq 23 where r is the number of AB bonds in the self-assembling macrocycle
EMnKinter g 185r
(23)
It is easy to show that also for the reaction of A-A + B-B the optimum monomer concentration for self-assembly is slightly more than one-tenth of nEMn, the cmc. By considerations analogous to those illustrated for the A-B reaction, one can obtain an approximate expression for the lsac which is valid when the acyclic fraction is composed of the monomers M1A and M1B only.
lsac )
1 1/(2n-1)
(2nEMn)
Kinter2n/(2n-1)
(24)
This occurs when x1/2/(1 - x1/2)2 ≈ x1/2, i.e., when x e 0.01. This in turn requires that
EMnKinter g (0.12/2n)102n
(25)
This condition is increasingly more demanding than the second condition for self-assembly, namely, eq 22, on going up from n > 2. Conclusion Two conditions are required for successful self-assembly macrocyclizations: (i) the self-assembling macrocycle must have an EM much larger than that of the other cyclic oligomers, as expressed more precisely by eq 6; (ii) the product EMnKinter must be no lower than 185r, where r is the number of bonds that hold together the monomer units in the self-assembling macrocycle, the higher the better. Thus, to design a self-assembling macrocycle, one should try to maximize the values of both EMn and Kinter; however, the product EMnKinter being equal, a large value of EMn is preferable to a large value of Kinter for the following reasons: (i) the value of EMn is important for the attainment of both the first and the second condition of self-assembly; (ii) a large EMn value implies that, apart from problems due to poor solubility of the equilibrating species, self-assembly can be carried out in more concentrated solutions (see the abscissa of the graph in Figure 2), thus increasing the efficiency of the process per volume of solvent; (iii) recovery from a kinetically controlled oligomer distribution toward equilibration is more rapid the lower the value of Kinter. Maximization of the EM requires careful design of the monomeric building block(s). The ideal building block is rigid (no built-in rotors) and has a structural predisposition to assemble in a unique cyclic oligomer without any strain. The ideal intermolecular reaction, also expressed in the structure of the building block, involves the formation of nonrotatable bonds. This can be realized by reactions involving either multipoints binding (for example multiple hydrogen bonding), or steric hindrance to rotation of the newly formed bonds, or the formation of multiple bonds (for example, reactions on transition metals involving back-donation, formation of imines, etc.). The ideal intermolecular reaction, of course, is also highly esoergonic in order to maximize the value of Kinter. However the selfassembly of large cyclic oligomers is hampered by two unavoidable factors, no matter how well the monomeric building block is designed. The first is due to the rotational symmetry of the cyclic oligomer that makes EMn decrease by a factor 1/r; this effect can be viewed as a consequence of the availability of r equivalent bonds for the ring-opening reaction of the cyclic product.3 The second is due to the fact that the required driving force for self-assembly, as expressed by the condition in eq 23, is directly proportional to r. The first factor is an obstacle for the attainment of both the first and the second condition of selfassembly. The latter, however can still be attained if both the decrease of EMn and the need for a larger driving force on increasing r are compensated by a sufficiently large value of Kinter. Provided that both the first and the second condition for selfassembly are met and an estimate of the EM of the selfassembling macrocycle is possible,4 the best monomer concentration to carry out the process is slightly more than one-tenth of the cmc () nEMn) (see eq 12). Deviation from this value is less and less important the higher the value of EMnKinter; however, the initial monomer concentration should not be lower than the lsac and higher than the cmc. Knowledge of both EMn
Physical Basis of Self-Assembly Macrocyclizations and Kinter allows the evaluation of the distribution curve of the self-assembling macrocycle as illustrated for the construction of Figure 2, thus permitting the choice of the most convenient initial monomer concentration. Acknowledgment. Thanks are due to Prof. L. Mandolini for his critical reading of the manuscript. References and Notes (1) For recent reviews, see: (a) Lehn, J.-M. Supramolecular Chemistry: Concepts and PerspectiVes; VCH Publishers: Weinheim, 1995. (b) Whitesides, G. M.; Simanek, E. E.; Mathias, J. P.; Seto, C. T.; Chin, D. N.; Mammen, M.; Gordon, D. M. Acc. Chem. Res. 1995, 28, 37-44. (c) Lawrence, D. S.; Jiang, T.; Levett, M. Chem. ReV. 1995, 95, 2229-2260. (d) Philp, D.; Stoddart, J. F. Angew. Chem., Int. Ed. Engl. 1996, 35, 11541196. (e) ComprehensiVe Supramolecular Chemistry, Atwood, J. L., Davies,
J. Phys. Chem. B, Vol. 102, No. 29, 1998 5703 J. E. D., MacNicol, D. D., Vo¨gtle F., Eds.; Pergamon: Oxford, 1996; Vol. 9. (2) Chi, X.; Guerin, A. J.; Haycock, R. A.; Hunter, C. A.; Sarson, L. D. J. Chem. Soc., Chem. Commun. 1995, 2563-2565. (3) Ercolani, G.; Mandolini, L.; Mencarelli, P.; Roelens, S. J. Am. Chem. Soc. 1993, 115, 3901-3908. (4) For a survey of the concept of EM and its applications to ring closure reactions, see: Mandolini, L. AdV. Phys. Org. Chem. 1986, 22, 1-111. (5) Jacobson, H.; Stockmayer, W. H. J. Chem. Phys. 1950, 18, 16001606. (6) The term predisposition has been used in this context to indicate a strong conformational or structural preference expressed by the building block once incorporated into a larger structure, giving rise to a thermodynamic preference for a particular product. Rowan, S. J.; Hamilton, D. G.; Brady, P. A.; Sanders, J. K. M. J. Am. Chem. Soc. 1997, 119, 2578-2579. (7) Margenau, H.; Murphy, G. M. The Mathematics of Physics and Chemistry, 2nd ed.; Van Nostrand Company: Princeton, 1962; p 492.
