Free Energy of Multiple Overlapping Chains
FFree Energy of Multiple Overlapping Chains
Katherine Klymko and Angelo Cacciuto ∗ Department of Chemistry, Columbia UniversityNew York, New York 10027 (Dated: November 4, 2018)How accurate is pair additivity in describing interactions between soft polymer-based nanopar-ticles? Using numerical simulations we compute the free energy cost required to overlap multiplechains in the same region of space, and provide a quantitative measure of the effectiveness of pairadditivity as a function of chain number and length. Our data suggest that pair additivity canindeed become quite inadequate as the chain density in the overlapping region increases. We alsoshow that even a scaling theory based on polymer confinement can only partially account for thecomplexity of the problem. In fact, we unveil and characterize an isotropic to star-polymer cross-overtaking place for large number of chains, and propose a revised scaling theory that better capturesthe physics of the problem.
The question of how nano and mesoscopic particlesspontaneously organize into complex macroscopic struc-tures can be considered as one of the great challengesin the field of soft matter today. In fact, the prospectof developing materials with new and exciting optical,mechanical, and electronic properties via the process ofself-assembly relies on our ability to predict and controlthe phase behavior of complex fluids.Although most of the work on self-assembly has his-torically focused on small molecules, the last decadehas witnessed several breakthroughs in particle synthe-sis at the meso-scale [1–5] making possible the produc-tion of nanoparticles with complex chemical and geomet-rical properties with an unprecedented degree of preci-sion. More recently, a burst of research activities has fo-cused on polymer-based nanoparticles [6]. What makesthese particles very interesting is that they break thedogma of mutual non-penetrability. Unlike regular col-loids, for which excluded volume interactions are strictlyenforced via a hard-core or a power-law potential, com-plex mesoparticles such as charged or neutral star poly-mers, dendrimers or microgels present a very peculiarpair potential describing their volume interactions. Be-yond the small and local deformation limit, which canbe easily described in terms of classical elasticity theory,these soft polymer-based nanoparticles have very unusualinteraction potentials which allow for even complete over-lap among themselves [6–8].Surprisingly, the simple relaxation of excluded volumeconstraints results in an extremely rich phase behaviorthat has been reported in several publications [8–17]. No-tably, it was found that some classes of soft interactionslead to reentrant melting transitions, others to polymor-phic cluster phases [18], and in general to multiple tran-sitions involving close-packed and non-close-packed crys-talline phases [19–21] as a function of the system den-sity. Remarkably, the phase behavior of these systemsis very much dependent on the the shape of the pair ∗ Electronic address: [email protected] potential, and it is today feasible to engineer interac-tions between star polymers or dendrimers by controllingtheir overall chemical/topological properties [22]. Likoset al. [23] established a criterion to predict whether fora bounded and repulsive potential re-entrant melting orcluster phases will occur based on the sign of the Fouriertransform of the interaction. For a recent review on thesubject we refer the reader to reference [6].Given the complexity of these nanoparticles, their in-teractions are usually extracted via an explicit coarse-grained procedure to reduce the problem to a simplerone consisting of single effective particles interacting viaan ad-hoc pair potential. Once a pair potential as a func-tion of separation r , F (2 , r ) is extracted, pair additivityamong any two effective particles is assumed and phasediagrams are computed.While it is by now clear that many body effects canlead to density-dependent interactions between poly-mer pairs [25], and methods to systematically includesuch deviations have recently been put forward [24, 26](and references therein), in this paper we show that thevery assumption of pair additivity can be greatly inad-equate in describing the interactions among polymer-based nanoparticles. To prove it, we compute the to-tal free energy cost associated with overlapping n p self-avoiding polymers (effective particles) of length N andshow that very quickly the assumed additivity of the pairinteractions breaks down.If one assumes straightforward pair additivity, the freeenergy cost required to overlap n p effective particles issimply proportional to the number of pair interactions βF ( n p ) = βF (2) n p ( n p − F ( n p ) by using the scal-ing theory based on polymer confinement as was first sug-gested by Jun et. al. [27]. The main idea is that in the di-lute limit overlapping two chains of length N , each havinga radius of gyration R G ∼ N ν , is equivalent to confininga single chain of length 2 N into a spherical cavity havinga radius equal to the radius of gyration of a single chain R G . The free energy cost associated with spherical con- a r X i v : . [ c ond - m a t . s o f t ] N ov finement of a polymer of length N into a spherical cavityof radius R is given by βF ∼ ( R G /R ) / (3 ν − (where β = ( k B T ) − ) [28–30]. By plugging in the denominator R → R G and in the numerator R G → (2 N ) ν , one obtains βF (2) ∼ ν/ (3 ν − . This equation, easily generalizableto n p chains as βF ( n p ) ∼ n ν/ (3 ν − p , states that (a) thefree energy cost to completely overlap two chains is in-dependent of their length (well established result boththeoretically and numerically [7, 8]), and (b) that over-lapping n p chains has a free energy cost that does notgrow linearly with the number of pair interactions.This scaling theory, that should better account forthe density increase within the overlapping region of thechains, can be re-written as βF ( n p ) = βF (2) (cid:18) n p − (cid:19) ν/ (3 ν − (2)For chains in a good solvent ν ≈ /
5, and the value of theexponent is close to 2 .
25 and is expected to cross over to3 for large number of polymers [27, 29]. To establish thecorrect dependence of the free energy with the numberof chains we performed numerical simulations.We modeled a chain as a sequence of N spherical beads(monomers) of diameter σ connected sequentially withan entropic spring of maximum extension √ σ . The in-teraction between any two monomers is described via ahard-core potential [31] while the entropic spring betweenconsecutive monomers has the form βV s ( r ) = (cid:40) σ < r < √ σ ∞ otherwise (3)The advantage of this model is that the free energy as-sociated with multiple overlapping chains is fully con-tained in their configurational entropy. To computethe free energy we used the thermodynamic integrationmethod [31]. The idea is to introduce a fictitious poten-tial βV i (cid:54) = jλ ( r ) = λ (cid:40) r > σ (1 − rσ ) if r ≤ σ (4)which acts exclusively among monomers associated todifferent chains ( i (cid:54) = j ), and to constrain the center ofmass of each chain to be within a spherical shell of ra-dius r = 2 σ . For λ = 0 the overlapping chains are notinteracting; as λ → ∞ we recover the system of interest.The free energy due to the polymer-polymer interactionscan then be extracted by performing the following inte-gral F ( n p ) = (cid:90) ∞ dλ (cid:18) dV λ dλ (cid:19) λ (5)In practice, we perform several Monte Carlo simulationsfor numerous values of λ until the hard sphere limit iseffectively reached, and compute numerically the inte-gral. In all our data the largest value of λ was selected n p F ( n p ) / F ( ) n p ( n p -1)/2 ( n p -2)/2)
3n /(3n -1)
Revised Theory N = 128 N = 64 FIG. 1: Free energy cost to overlap n p chains as afunction of the number of chains. The dashed anddotted lines represent respectively the straightforwardand the scaling-based theory from Eq. 1 and Eq. 2respectively. The solid line shows the revised theory inEq. 6. The inset is a zoom of the data for small n p .to be the one for which the total energy of the systemwas decreased to a value of the order of 10 − . The smalltail of the integral for larger values of lambda was com-puted by fitting the tail of the data with a power lawand by extending the integral until the change of F ( n p )becomes clearly negligible (this typically accounts for avery small amount of the total free energy). To properlysample V λ our simulations were run from a minimum of50 million to a maximum of 250 million sweeps. To im-prove our statistics, after every system sweep a globalchain rotation move is also implemented. This consistsof randomly picking a monomer and a direction in a ran-dom chain, and of performing a rigid rotation around arandomly selected axis of a small random angle of all themonomers connecting the selected monomer to the endof the chain along the selected direction. Our simula-tion were performed in the N V T ensemble in the dilutelimit ( V (cid:29) / πR ) for polymers of length N = 64 and N = 128, with n p ranging from 2 to 32.Our results indicate that the free energy required tosuperimpose two chains equals βF (2) = 2 . N = 128 and βF (2) = 2 . N = 64. These resultsare fully consistent with previous numerical simulationson similar systems, and the small difference is simply dueto finite size effects. Figure 1 presents the core results ofour simulations and shows how F ( n p ) normalized by thepair free energy F (2) varies with the number of poly-mers. Lines indicating the straightforward (Eq. 1) andthe scaling-based (Eq. 2) predictions are also shown asa reference. Our findings clearly show that Eq. 1 be-comes inaccurate as soon as n p >
4, Eq 2 is accurateup to n p ≤ n p . Interestingly, the dependence of the freeenergy with the number of chains becomes weaker as thenumber of chains increases. n p R G N = 128 N =64Fit to the data FIG. 2: Average radius of gyration ¯ R G of multipleoverlapping chains as a function of the number of chains n p . The line is a fit to the data.This result is a bit counter intuitive because one shouldexpect a stronger dependence of F ( n p ) on n p as thepolymer concentration is increased inside the overlappingarea. The key issue with the scaling theory is that it as-sumes that the size of the confining cavity (which equalsthe radius of gyration of an unconstrained polymer R G )remains constant for any number of polymers. Althoughthis is a good assumption for few chains, we find thatas n p increases the average size of the confining cavitybecomes systematically larger. This is because chainsdo swell to minimize the number of interactions. Fig-ure 2 shows the average size of the system, measured bycomputing the average radius of gyration of the chainsat large values of λ , (cid:104) R G ( n p ) | λ = ∞ (cid:105) , divided by the ra-dius of gyration of the reference non-interacting poly-mers, (cid:104) R G ( n p ) | λ =0 (cid:105) . By fitting these data to a powerlaw ¯ R G ( n p ) = a + b n αp , we can explicitly account for thiscorrection. The revised free energy scales as βF R ( n p ) ∼ (cid:0) n νp / ( a + b n αp ) (cid:1) / (3 ν − (6)with a (cid:39) . b (cid:39) . α (cid:39) .
21. As seen in Fig. 1, thisfunction successfully fits most of our numerical data upto n p = 16. Crucially, the scaling behavior of the radiusof gyration with the number of chains α ≈ / n / p (cid:28) N ). This suggests that a rearrangement fromisotropically mixing chains to demixed/localized chains(star-polymer) may be taking place. To investigate thisscenario, we computed the asphericity of the polymers asa function of n p . This is obtained by computing the in-ertia tensor of each polymer, and by combining the threeeigenvalues l , l and l into the rotational invariant pa-rameter [32] q = ( l − l ) + ( l − l ) + ( l − l ) l + l + l ) (7)The plot of the normalized asphericity ¯ q = q ( n p ) /q (1) aswell as the three normalized eigenvalues ¯ l i = l i ( n p ) /l i (1) n p q N = 64 N = 128 n p l i i = 1 i = 2 i = 3 FIG. 3: Average asphericity ¯ q for N = 64 and N = 128, computed according to Eq. 7, for multipleoverlapping chains as a function of the number of chains n p . The inset shows the three normalized eigenvalues ofthe shape tensor ¯ l i for N = 64.as a function of the number of polymers n p is shownin Fig. 3. For a polymer in a good solvent a first orderepsilon expansion results, in the dilute limit and for N →∞ , to a value of q close to 0.415. Accurate numericalsimulations predicted a value closer to 0.431 [33, 34]. Forour longest chains, we find that indeed q = 0 . q increases significantly to more than 1.6 times its original.This is a clear indication that the chains are not swellingisotropically as n p increases, but are stretching out alongtheir long axis at expenses of the other two directions.This morphological change of the chains is suggestive ofthe fact that chain segregation may also be occurring.To check for chain segregation we tracked the locationof the chains’ main axes over the course of a long sim-ulation trajectory, and projected it over the surface ofa sphere centered around the system. Figure 4 tracksdown the locations of polymers’ axes for n p = 2 and n p = 32. For the sake of clarity we show the tracks ofonly two randomly selected chains also for the case of n p = 32. Clearly, when n p = 2 the chain’s axes canexplore the whole spherical surface, indicating completemixing of the chains. In the latter case only a small re-gion of the surface is explored by the two selected chains,clear indication of chain segregation.All of our data points to the fact that our system iseffectively behaving as a star-polymer, i.e. a system ofpolymers radiating from a central core, and segregatedinto roughly conical regions. A power law fit of our dataincluding only values of the free energy for n p ≥
15 resultsin βF ( n p ) ∝ n . p which is indeed compatible with starpolymers in the semi-dilute limit [35].In summary, we computed the free energy cost associ-ated with the complete overlap of multiple self-avoidingchains as a function of chain number. Our data show thatalthough the free energy error associated with pair addi-tivity of potentials between soft polymer-based nanopar-FIG. 4: Chains’s axial maps. The l.h.s tracks thelocation of the main axis of two chains projected onto aspherical surface centered around the system over thecourse of a long Monte Carlo trajectory when only twochains are present. The r.h.s shows the same map fortwo randomly selected chains in a system containing 32overlapping polymers. In the former case the polymers’axes perform a random walk over the spherical surface,indicating complete mixing of the chains, whereas in thelatter case the chains are clearly segregated in specificregions.ticles is of the order of just a few k B T for n p ≤ n p . We have alsoshown that an extended scaling theory based on poly- mer confinement of a single chain into a spherical cavitycan better account for the free energy cost of multipleoverlapping polymers. Finally, we have shown that de-mixing and chain segregation occurs when consideringlarge numbers of chains, and that in this limit, the freeenergy of a star-polymer provides a better description ofour numerical data.Although our study focuses on single polymer chains,the main message should also hold for multi-polymer-based particles such as star-polymers and dendrimers,where deviations could be much more significant even forsmall n p and could have dramatic consequences especiallyfor the phase diagrams associated with systems formingcluster crystals where multiple particles overlap on thesame lattice sites. It should be stressed that our resultscan be directly linked to the chain segregation processdescribed by Jun et. al. [27] for polymers under confine-ment. In this analogy, the formation of a star-polymercan be understood as chain segregation with the addedconstraint of overlapping center of masses. ACKNOWLEDGMENTS
This work was supported by the National ScienceFoundation under CAREER Grant No. DMR-0846426. [1] DeVries et al., Science , 358 (2007).[2] M. Li, H. Schnablegger, and S. Mann, Nature , 393(1999).[3] L. Hong, S. Jiang, and S. Granick, Langmuir , 9495(2006).[4] H. Weller, Phil. Trans. R. Soc. A , 229 (2003).[5] E. K. Hobbie et al., Langmuir , 10284 (2005).[6] C. N. Likos, Soft Matter , 478 (2006).[7] A. Y. Grosberg, P. G. Khalatur, and A. R. Khokhlov,Makromol. Chem. Rapid Commun. , 709 (1982).[8] A. A. Louis, P. G. Bolhuis, J. P. Hansen, and E. J. Meijer,Phys. Rev. Lett. , 2522 (2000).[9] O. G¨otze, H. M. Harreis, and C. N. Likos, J. Chem. Phys. , 7761 (2004).[10] C. N.Likos, H. L. Watzlawek, B. Abbas, O. Jucknischke,J. Allgaier, and D. Richter, Phys. Rev. Lett. , 4450(1998).[11] A. Jusufi, C. N. Likos, and H. L¨owen, Phys. Rev. Lett. , 018301 (2002).[12] A. R. Denton, Phys. Rev. E , 11804 (2003).[13] D. Gottwald, C. N. Likos, G. Kahl, and H. L¨owen, Phys.Rev. Lett. , 68301 (2004).[14] C. Pierleoni, C. Addison, J. P. Hansen, and V. Krakovi-ack, Phys. Rev. Lett. , 128302 (2006).[15] B. Capone et al., J. Phys. Chem. B , 3629 (2008).[16] B. Bozorgui, M. Sen, W. L. Miller, J. C. P´amies, andA. Cacciuto, J. Chem. Phys. , 014901 (2010).[17] A. Cacciuto and E. Luijten, Nanoletters , 901 (2006).[18] B. M. Mladek, D. Gottwald, G. Kahl, M. Neumann, and C. N. Likos, Phys. Rev. Lett. , 045701 (2006).[19] A. Suto, Phys. Rev. B , 104117 (2006).[20] C. N. Likos, Nature , 433 (2006).[21] J. Pamies, A. Cacciuto, and D. Frenkel, J. Chem. Phys. , 044514 (2009).[22] B. M. Mladek, H. Kahl, and C. N. Likos, Phys. Rev. Lett. , 028301 (2008).[23] C. N. Likos, N. Hoffmann, and H. L¨owen, Phys. Rev. E , 031206 (2001).[24] I. Coluzza, B. Capone, and J. P. Hansen, Soft Matter ,5255 (2011).[25] P. G. Bolhuis, A. A. Louis, and J. P. Hansen, Phys. Rev.E. , 021801 (2001).[26] A. J. Clark and M. G. Guenza, J. Chem. Phys. ,044902 (2010).[27] S. Jun and A. Arnold, Phys. Rev. Lett. , 128303(2007).[28] A. Y. Grosberg and A. R. Khokhlov, Statistical Physicsof Macromolecules (American Institute of Physics, NewYork, 1994).[29] W. L. Miller and A. Cacciuto, Phys. Rev. E , 021404(2009).[30] T. Sakaue and E. Rapha¨el, Macromolecules , 2621(2006).[31] D. Frenkel and B. Smit, Understanding Molecular Sim-ulation: From Algorithms to Applications (AcademicPress, London, 2002).[32] J. Rudnick and G. Gaspari, J. Phys. A: Math. Gen. ,L191 (1986). [33] O. Jagodzinski, E. Eisenriegler, and K. Kremer, J. Phys.I (France) , 2243 (1992).[34] J. W. Cannon, J. A. Aronovitz, and P. Goldbart, J. Phys. I (France) , 629 (1991).[35] M. Daoud and J. Cotton, J. Phys. (Paris)43