Entropy Driven Self-Assembly in Charged LockâKey Particles

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Entropy Driven Self-Assembly in Charged Lock-Key Particles Gerardo Odriozola, and Marcelo Lozada-Cassou J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b01805 • Publication Date (Web): 31 Mar 2016 Downloaded from http://pubs.acs.org on April 1, 2016

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The Journal of Physical Chemistry

Entropy Driven Self-Assembly in Charged Lock-Key Particles Gerardo Odriozola† and Marcelo Lozada-Cassou∗,‡ Área de Física de Procesos Irreversibles, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Av. San Pablo 180 Col. Reynosa, 02200 México D.F., Mexico, and Instituto de Energías Renovables, Universidad Nacional Autónoma de México (U.N.A.M.), 62580 Temixco, Morelos, México. E-mail: [email protected]

To whom correspondence should be addressed Área de Física de Procesos Irreversibles, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Av. San Pablo 180 Col. Reynosa, 02200 México D.F., Mexico ‡ Instituto de Energías Renovables, Universidad Nacional Autónoma de México (U.N.A.M.), 62580 Temixco, Morelos, México. ∗ †

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Abstract In this work we study the Lock-Key model successfully used in supramolecular chemistry and particles self-assembly and gain further insight into the infinite diluted limit of the lock and key, depletant mediated, effective attraction. We discuss the depletant forces and entropy approaches to self-assembly and give details on the different contributions to the net force for a charged lock and key pair immersed in a solvent plus a primitive model electrolyte. We show a strong correlation of the force components behavior and the underlying processes of co-ion and solvent release from the cavity. In addition, we put into context the universal behavior observed for the energy-distance curves when changing the lock and key to solvent size ratio. Basically, we now show that this behavior is not always achieved and depend on the particular system geometry. Finally, we present a qualitative good agreement with experiments when changing the electrolyte concentration, valence, and cavity-key size ratio.

1. INTRODUCTION Close to a decade before the establishment of the existence of atoms and molecules as something more than a convenient theoretical constructs, Emil Fischer was disentangling the stereo-chemical and isomeric nature of sugars 1 and proposing the "Lock and Key Model" to uncover the substrate-enzyme selectivity of enzyme-catalyzed reactions. 2 Much later, Asakura and Oosawa (AO) proposed their model of colloidal attraction by depletion . 3,4 In their view, a depletion effective force between large particles arises as a consequence of the appearance of a depletion region of smaller ones between them. These two ideas were recently combined to produce the self-assembly of lock and key colloids, 5 which is nowadays recognized as one of the possible and sound strategies for the bottom-up approach of the developing of new materials. Consequently, a great deal of attention has been recently paid to such systems, 6–14 which goes from their pair effective attraction, 7,15–19 to their phase behavior, 18–21 passing trough their binding kinetics 22,23 and self-assembly under confinement. 13 2


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On the other hand, the comprehension of self-assembly seems to be also relevant at the molecular level (by simply appealing to the frcuently found colloidal-atomistic analogy), such as for the deoxyribonucleic acid (DNA) complexation, 24–32 DNA self-assembly 33 and viral self-assembly, 34–41 in spite of the strong electrical repulsive forces involved. Lock and key particles present geometrical complementarity, i. e., the pair is anisotropic and have at least a concave surface (on the lock) where the key matches. For hard bodies, shape anisotropy is known to lead to anisotropic forces and to produce richer phase diagrams than those having an isotropic nature. 42 In addition, the presence of concave surfaces make these forces to enhance their strength in the “complementary direction” (directional binding 11 ) and to produce even more complex phase diagrams where unusual phases are frequently found. 11,20,21 Despite the fact that these phases are dictated by the solely shape of their components (for hard systems), their prediction is not possible without the aid of statistical mechanics. Thus, the exact way the shape of the particles affects the equilibrium structure of the whole is still not completely understood. 43–45 To aid having a clearer view, it is useful to comprehend the effective interaction of a couple of macro-particles immersed in a sea of smaller ones. In the past we have approach to this problem through model hard systems, 15 and model hard systems plus a Coulomb interaction. 17 In these works we have shown the possible routes for binding, 15 how there exist a universal behavior of the effective attractive potential when changing the ratio between the macroparticles size and the solvent size, 17 and how the lock-key entropy rules the interaction despite the presence of a relatively large and direct Coulombic repulsion. 17 A mechanistic explanation, based on depletion forces, as proposed in the AO model, of the self-assembling of colloidal particles, is probably not satisfactory. Even beyond the well known limitations of the AO model theories, 46–49 colloidal particles in a colloidal dispersion are hundreds or thousands of Angstroms apart, i.e., much beyond these particles inter-molecular forces range (London-van der Waals, electrostatic forces, etc.), 44 which are as much as a few Angstroms, in non-charged systems, or a few hundred Angstroms in, for

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example, highly charged colloidal dispersions. 44 Hence, concepts as molecular recognition, often referred to in relation with colloidal self-assembly 50 or supra-molecular chemistry, 43,51,52 should probably be better understood in terms of the entropy interpretation of the missing information, as defined in the Information Theory, 53–56 i.e., self-assembling is ruled by entropy, as has been pointed out in the past, 15,17 and now widely recognized. 7,9,13,14,22,45,57,58 This entropy mechanism, behind the long range particles correlation, or recognition, is exacerbated when large paricles, at finite concentration are considered. 59 The depletion forces vs. entropy approach to self-assembling resembles that of the vis viva controversy between the conservation of momentum (Newton-Descartes) vs. the conservation of energy (Leibnitz). We now know that these two approaches are complementary. Self-assembling, however is responding to the system needs to distribute its energy (entropy), something that goes beyond a force valance approach. In this work we give further details of our previous results and perform extra calculations on the effective interaction of charged lock-key particles in a model electrolyte. In particular we are now considering different types of electrolytes (1-1 and 2-2), modifying their concentration, and analyzing the case where the key and lock particles have opposite sign. We are also showing the limitations of an energy vs. lock-key distance master curve, presented in our previous Monte Carlo (MC)simulations studies 17 . For this purpose we have included new data series for uncharged lock and key pairs, where the key does not perfectly match the cavity size, and the key to solvent size ratio is varied. The paper is organized as follows. Following this brief introduction, in the next section we present the models and the method employed in our MC calculations. Immediately after results are described. This section is split in three parts. The first deals with charged lock and key macro-particles immersed in a 1-1 electrolyte, restricted primitive model solution, plus solvent particles, taken as hard spheres. The second give further details on when the universal scaling behavior applies for increasing the macro-particles to solvent size ratio. The third subsection is devoted to theoretically reproduce some of the experimental findings

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given in Sacanna et al., 5 involving the change of the lock and key match, their concentration, and the type of electrolyte employed for screening the direct macro-particle’s charge. Finally some conclusions are presented.

2. MODEL AND METHOD

Figure 1: Macro-molecules (blue glass texture) at the distance of lower free energy for the perfect lock-key match, σc = σk , for a key particle eight times larger than the solvent, σk = 8σs , an electrolyte concentration of 0.15 M, and for likely and highly charged lock and key particles, zl = zk = 32. Red and blue particles represent the counter and co-ions, respectively. Counter-ions are seen inside the lock partially screening the direct electrostatic interaction. Solvent particles are shown as dots to gain clarity. Even for this highly charged system, the assembled configuration is the one with the lower free energy due to entropy gained by the solvent (there is a large increase of the solvent accessible volume when the lock-key pair is bonded). The volume fraction is ϕ = 0.2. We employed standard NVT Monte Carlo (MC) simulations 60 to study a system counting on a lock-key pair, electrolyte (when considering charged lock and key particles), and solvent. A hard sphere of diameter σl which has a spherical cavity of diameter σc = σl /1.7 located p at a distance d = 1/2 σl2 − σc2 from its center accounts for the hard contribution of the lock particle. The key hard core contribution is taken into account by a hard sphere of diameter σk which, unless otherwise stated, perfectly matches the lock cavity, i. e. σk = σc . The macro-molecules pair is fixed during a whole MC run and located with their centers 5



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(including the cavity center) over the x-axis, in such a way that the cavity and the key particle are faced. We are labeling as x to the distance between the lock-key positions, taking x = 0 at contact. Solvent particles are simply hard spheres with a fixed diameter, σs = 3.0Å. In case of studying the influence of charges in the system, we set a point charge at both lock and key centers with the same absolute valence |zl | = |zk |. In this case, we also added two more species, cations and anions, by means of a restricted primitive model (RPM) electrolyte with σe = 4.25Å. The whole system is forced to be electroneutral by adding the necessary cations or anions. The electrostatic interaction is given by

UE (rij ) =

ℓB zi zj , βrij

(1)

being β=1/kB T , kB the Boltzmann constant, T =298K the absolute temperature, zi and 2

zj the valences of sites i and j, ℓB = βeǫ =7.14Å the Bjerrum length, ǫ = 78.5 the (water) dielectric constant, and rij the inter-site distance. For simplicity, lock-key particles and solution are given the same dielectric constant. Periodical boundary conditions are set for the three orthogonal directions and the Ewald summation formalism is employed to deal with the electrostatic interaction. To allow for ion interchange between the confined and unconfined regions (inside - outside a partially covered lock cavity), we account for movements having a large maximum displacement. This is, in other words, to guaranty a constant chemical potential for solvent and electrolyte species on the whole system. A snapshot of an equilibrium configuration corresponding to the system with σc = σk , σk = 8σs , zl = zk = 32, an electrolyte concentration of 0.15 M, and a total occupied volume fraction (key-lock, ions and solvent particles) of ϕ = 0.2 is shown in Fig. 1. In this snapshot the macroparticles are set approximately at the distance of minimum free energy. The simulation box has sides lengths Lx = min[σl + σk + 12σs , 25σs ], Ly = Lz = min[σl + 6σs , 14σs ]. These conditions are sufficient to avoid size effects. The origin of coordinates is set at the box center. In all of our simulations ϕ = 0.2, unless otherwise indicated. Hence, for different

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values of σk , σl and/or electrolyte concentration, the box size and solvent concentration will be adjusted accordingly, to keep the reported total volume, fraction, ϕ, constant. Notice that σk , σl and σc , are all scaled with σs . Electrostatic contributions to the forces acting on macro-particle i are obtained by

Fel i = h

X

−∇UE (rij )i,

(2)

j

where i is the macro-molecule charged site and j runs over all other charged sites. The contact (depletion or entropy) contribution is obtained by integrating the ions and solvent contact concentration, ρ(s), Fci

= −kB T

Z

ρ(s)nds,

(3)

s

where s refers to the macro-particle surface and n is a unit vector, normal to the surface, and pointing towards the fluid. To correctly evaluate equation 3 a careful extrapolation of the fluid concentration towards the surface is needed. We employed kB T /Å and kB T units for forces and energies, respectively. These two contributions to the overall force are interdependent. Of course, the third Newton’s law must be fulfilled by the average forces acting on the lock and key particles. Thus, no averaged net forces act on the pair as a whole, and on the fluid (with and without considering the macro-particles). These facts are employed to check the correctness of the implementation. Once the forces for several macro-particles separation distances are obtained, the energy is calculated by integration, Rx i. e., by means of E(x) = ∞ Fx dx. Here Fx is the x component of the force (symmetry around the x-axis forces the other Cartesian components to vanish).

3. RESULTS This section is split into three parts. In the first one we deal with charged macro-particles immersed in a 0.15M restricted primitive model 1-1 electrolyte solution, plus a hard spheres

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solvent. This is done with the aim of directly compare the electrostatic (enthalpy) component of the energy with the depletion contributions from solvent for several cases. In the second part, we restrict the analysis to the uncharged system while varying the shape of the involved particles. In this section we focus on in which circumstances the energy as a function of distance can be represented by a the master curve. Finally in the last part, we show how factors as solvent concentration, valence of the electrolyte, electrolyte concentration, cavitykey relative size, and lock-key concentration control the fraction of assembled pairs.

3.1. Charged macro-particles in a 1-1 electrolyte solution In Fig. 2 we give an example on how the different contributions to forces and energies behave with distance. This figure shows on the left panel the different contributions to forces, and on Rx the right the corresponding energies, following E c (x) = ∞ Fxc dx, where super-index c refers to a given contribution. These are the force components coming from solvent, counter-ion, and co-ion, surface-contact concentrations entering equation 3 and the total electrostatic contribution from equation 2. For this case, we set an electrolyte concentration of 0.15M , a relative size of the lock-key pair with respect to solvent given by σk /σs =6, and highly charged macro-molecules, zl = zk =16. In the case of Fig. 2, ϕ = 0.2 implies a solvent bulk concentration ρs = 19.2M . However, as pointed out above, for other values of σk /σs or electrolyte concentrations, ρs will be adjusted, along with the simulation box size, to have the reported ϕ. For large separation distances all contributions to forces vanish, as expected. When decreasing the distance between macro-particles the different contributions build up, with the only exception of the coion contact force which is always zero. This is also expected for highly charged particles showing strongly polarized double layers, which produce a relatively large excluded volume region of co-ions near the macro-particle surface. Thus, co-ions rarely touch macroparticles producing no contact forces. 30 Conversely, counter-ions largely adsorb on macro-particles leading to forces when the adsorption symmetry is broken. This 8


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is the case corresponding to Fig. 2. The left panel shows a large repulsive peak close to 6Å (squares). At this distance counter-ions are mostly attached at the inter-macro-particle region filling the lock cavity. In this way they produce an additional lowering of the free energy trough enthalpy (this enthalpy-driven adsorption corresponds also to the case shown in Fig. 1, although for different parameter values). Here, a further decrease of the inter-particle distance leaves no room for ions inside the cavity, explaining the abrupt change of sign of the counter-ion contact contribution to the force. Finally, a practically constant attractive counter-ion contribution is observed for small distances, due to the approximately invariant external pressure they yield on the pair. The small slope can be explained in terms of an increase of the effective charge surface density ((zl + zk )/Alk , being Alk the lock and key area exposed to the fluid) of the lock-key pair for decreasing distances and the larger counter-ion surface concentration they produce (not shown).

Figure 2: Force (a) and energy (b) contributions to the lock-key total force and total energy, respectively, for a system with σk /σs =6 and for zl = zk =16, as a function of the distance between the inner surface of the lock cavity to key surface, x. The inset zooms in the energy data. The attractive (negative) contact contribution from solvent molecules and the repulsive (positive) electrostatic contribution are clearly dominant. Counter-ions also produce a relatively small contact and repulsive contribution while screening the direct electrostatic force. Co-ions play no practical role. Note that a relatively large lock-key charge (zl = zk =16) is needed to counterbalance the collective attractive force. ϕ = 0.2. The other two important contributions are closely related to the counter-ion release

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process. 31 These are the total electrostatic component (triangles of Fig. 2) and the contact force from solvent molecules (circles of Fig 2). The link between electrostatic and counter-ion distribution, is clearly observed in panel (a) of Fig. 2. Upon reaching the maximum of the counter-ion contact contribution and when a further decrease of distance leads to a counterion release from the cavity, the screening turns poorer which causes the total electrostatic force to steeply increase. On the other hand, the solvent component is correlated to that of counter-ions due to excluded volume effects. That is, a large counter-ion concentration inside the cavity necessarily provokes the exclusion of solvent molecules from this region. Thus, the solvent contribution to the force is always attractive and stronger for decreasing distances, but it reaches a plateau when counter-ions are released from the cavity. This plateau occurs due to the entering of solvent molecules to the cavity, as they replace the co-ions. A further decrease of the lock-key distance leads to a force jump at approximately 4Å. This jump, as for counter-ions, results from the abrupt release of solvent molecules from the cavity. Once the cavity runs out of solvent molecules as decreasing the macro-molecules distance, the solvent force contribution turns a constant. Conversely to the counter-ions, the surface concentration of uncharged particles does not depend on the effective charge surface density of the lock-key pair. As a consequence of the steep counter-ion and solvent two-step release from the cavity, the overall force (triangle-down symbols of Fig. 2) jumps from repulsive to attractive two times in the small range of 4 − 10Å. This behavior can be even much more complex when considering partially hydrated ions (we are assuming ions and their first hydration shell to behave as a rigid sphere), and/or a charge distribution on the macro-molecules surfaces and their corresponding (complete or partial) hydration shells. These effects, among others, can be important when inter-particle distances are small. On the other hand, the integrals Rx E c (x) = ∞ Fxc dx are, in general, soft and well behaved as observed in panel (b) of Fig. 2.

Indeed, this behavior is highlighted in the inset, where the same data are zoomed in. There it is observed an always repulsive contribution of counter-ions (contact component), an

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Figure 3: Total energy as a function of as a function of the distance between the inner surface of the lock cavity to key surface, x, with σk /σs as parameter (as shown in the figure label). Panels a, b, c, and d correspond to zl = −zk = 1, zl = zk = 1, zl = zk = 2, and zl = zk with ezk /(πσk2 ) = 0.2833 C/m2 , respectively. ϕ = 0.2. ever attractive contribution of solvent, an always repulsive electrostatic contribution, and a double-minima total energy (secondary minima is shallow in this case) split by a O(kB T ) barrier which resembles the DLVO potential of large hydrophobic colloids. As shown further in the text, a repulsive total energy at the lock-key contact is found only when relatively large direct electrostatic forces are set. Thus, the double-minima energy is not general, but a consequence of the particular choice of the parameters. A final detail to note is the existence of a tiny attractive co-ion contribution (see the inset of panel (b) of Fig. 2). Putting aside the several components to the total energy and the underlying details of the mechanisms leading to it, we now focus attention on how the total energy looks like by varying the relative size ratio σk /σs and the charges on the macro-particles. We are considering the relative size ratios σk /σs =1,2,4,6 and 8 (different curves for all panels of Fig. 3), and the cases zl = −zk = 1, zl = zk = 1, zl = zk = 2, and zl = zk with zk /(πσk2 ) = 0.2833 C/m2 (panels a, b, c, and d of Fig. 3, respectively). Panel (a) presents results for macro-particles with 11



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opposite sign. In this case the main contributions, contact from solvent and electrostatic, are attractive (not shown). Furthermore, there is a small release of ions from the inter-particle region which also produces attractive depletion contributions (not shown). Thus, binding energies (lock-key at contact) are large and increase with increasing macro-particle-solvent size ratio σk /σs . Nonetheless, it is also observed the development of a barrier which shifts to larger distances with increasing σk /σs . This barrier is ruled by the solvent contribution alone and is related to the higher concentration of the solvent molecules, in between the lock and key particles, when they are sufficiently apart from each other, previous to its release from the cavity, when they approach to each other, to make possible the lock-key binding. As was shown in previous work, it is possible to avoid this kinetic barrier by allowing the key to touch the cavity border before producing the match. 15 In fact, the electrostatic contribution decreases with increasing σk /σs , as expected for larger macro-particles with a fixed central charge. On the other hand, the entropy contribution turns larger with larger σk /σs , which is also expected due to the size of the cavity enlargement. Hence, the relative electrostatic contribution to the overall energy steeply decreases. The small effect of the electrostatic contribution turns evident when comparing panel (a) with panel (b), where we switched the attractive electrostatic contribution to repulsive, i. e., we changed from zl = −zk = 1 to zl = zk = 1, respectively. The practically negligible electrostatic contribution is clear for cases σk /σs ≥ 4. However, for small macro-particles, electrostatic contributions are not negligible. Indeed, it turns important for the (somewhat unrealistic) case σk /σs = 1, where the solvent depletion and the electrostatic component are comparable and with opposite sign (not shown). This give rises to the peculiar shape of E(x) in this case. Further proof of the little impact of the electrostatics on the lock-key binding process for σk /σs ≥ 4 is shown in panel (c) of the same figure. Here, we set zl = zk = 2 and nearly identical curves result for relatively large lock and key particles (compare diamonds, triangles, and triangles-down symbols of panels (a) to (c)). Again, a different story can be tell for macro-particles with a size comparable to that of solvent molecules. In fact, electrostatic

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forces rule for σk /σs = 1. Conversely to the constant charge case, setting a constant surface charge density on the key particle, for every σk /σs ratio, which implies increasing zk for increasing σk , such as ezk /(πσk2 ) = 0.2833 C/m2 , where e is the proton charge, and correspondingly increasing zl such that zl = zk , give rises to a relative increase of the electrostatic forces compared to the depletion forces for the matching (contact) distance with increasing macro-particles size, for σk /σs ≥ 2. The increase of the repulsion at contact is observed in panel (d) of Fig. 3 for all curves except circles σk /σs = 1, and and squares (σk /σs = 2). The circles and squares curves in panel (d) correspond to a zl = zk ≈ 0.5 and zl = zk ≈ 2, respectively, which explains why the circles curve in panel (d) has a lower contact value than the circles curve in panel (b), and the squares curve of panel (d) almost perfectly matches the squares curve of panel (c). For larger distances, where counter-ions can enter the cavity, the contact contribution from counter-ions turn comparable to the pure electrostatic term. Thus, both, contact from counter-ions and electrostatic contributions are important and opposes the solvent depletion (co-ions play no practical role). The sum of all contributions results in a curve showing a low energy minima such that the lock and key assembly would result in a loose configuration, letting counter-ions rest inside the lock cavity and directly screen the naked macro-particles charges. This low energy configuration is the one shown in Fig. 1 (macro-particles are given a glassy texture to show counter-ions inside the cavity). For larger distances, again, a kinetic barrier develops, which is of the order of kB T . In all panels of Fig. 3 there is a relatively "long range" energy minimum between the lock and key particles, which is, of course, a result of an effective long-range many-body force, and can not be explained in terms of simply depletion forces models, even for more refined models, beyond the AO model, where the solvent ( or polymer) molecules are taken as an ideal (volume-less) gas having size only at contact with the colloid surface. In summary, the entropy driven lock-key interactions favor a contact or long range lock-key pairing, even if strong repulsive Coulomb forces are present.

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master curve can be obtained following the same procedure, i. e. by normalizing energy and distance with Emax and xmax (whose values are shown in panel (b) of Fig. 5). Also, a similar shape is kept. That is, it shows a deep energy minimum at contact (approx. 7.5E/Emax ), a maximum and a series of oscillations around zero of decreasing amplitude for increasing distance. Nonetheless, both master curves are clearly different, and so, universality is kept only when the lock-key geometry is kept invariant. This result was expected since, for example, a donut-like lock and its corresponding matching key would produce, due to symmetry arguments, a null force at the matching position. This is a strong difference with our lockkey pair which produces the maximum attractive force at contact. Thus, universality cannot generally apply to different lock-key shapes. In fact, universality for scaling the lock and key pair is not even a rule for a given key-lock geometry. An example for a system clearly braking the scaling is given in panel (c) of Fig. 4. In this case we set a smaller key than the lock cavity, σk /σc = 0.7. As a consequence, solvent particles can enter, or not, to join the key inside the cavity, depending on their relative size. When there is room for them along with the key inside the cavity, the attractive energy at contact drastically decreases (approximately | −2.0E/Emax |). This is observed for cases σk /σs ≥ 6. When there is no room for the key and the solvent molecules inside the cavity, the energy at contact is close to −8.0E/Emax (similar to the case of panel (b)). This steep change for the contact energy can be employed by means of a conformation change of the lock to produce a desired release of a given substrate, as previously suggested. 15 In brief, for key particles smaller than the cavity enlarging the σk /σs ratio produces curves which cannot collapse to define a single one. Of course, other geometries may behave similarly. Panels (a) and (b) of Fig. 5 show Emax and xmax for the cases where scaling was observed, i. e. for cases of panels (a) and (b) of Fig. 4, respectively. In both cases Emax and xmax show a similar behavior, i. e. they increase with increasing σk /σs . Hence, an increase of the macro-particles size relative to the solvent leads to an increase of the magnitude and range of the attraction energy, since an increase of the macro-particles size increases the

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Figure 5: Energy maximum and corresponding distance given in units of σs and σk for cases (a) and (b) of figure 3. Both, magnitude and range increase with increasing macromolecules-solvent size ratio. Thus, increasing the relative size between macro-molecules and solvent favors attraction. However, the range increases at a lower rate than the relative size, and thus, the attraction range, in units of key diameters, decreases.

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available volume, thus the entropy. Hence, given a constant repulsive contribution of any nature, increasing the relative size of macro-particles favors binding. This, in our view, may explain why enzyme-substrate complex formation is so frequently found in biochemical systems. On the other hand, the range of the interaction given in units of the macro-particles sizes decreases with their relative size to the solvent, despite the fact that the absolute range increases. This may explain why certain type of micron-sized lock and key colloids suspended in water do not bind unless a relatively large, non-adsorbing, water-soluble depleting agent is added. 5 In other words, the contact force interaction range is strongly increased by adding the polymer. Nonetheless, surfaces separated by not too many water molecules are certainly influenced by the contact water interaction, which, in line with van der Waals forces, leads to irreversible aggregation. 61 The data given in Fig. 4, on the one hand, show that the relative size of the macro-particles with respect to the solvent must be large enough to produce a sufficient increase of the solvent degrees of freedom to overcompensate the entropy lose due to the macro-particles assembly. On the other hand, a very large relative size produces a short range interaction, in units of the macro-particles size, of practically irreversible nature. This suggest the existence of lower and upper limits for the relative size of the macro-particles with respect to the depleting agent for the complexation phenomena for all scales (proteins-solvent or colloids-polymers).

3.3. Controlling the fraction of assembled pairs This section is focused on qualitatively reproducing the experimental data of lock and key colloids. 5 Prior to the formation of reversible complexes, lock and key colloids are stabilized by surface charge groups and/or by sterical repulsion. As mentioned, the range of these interactions are far beyond the range of the solvent (water) contact contribution, and thus, a large depleting agent is added to extend its range which is linked to the depleting gyration radius. Hence, for computation purposes, the media (water) is considered as a continuous space and our hard spheres takes the place of the depleting agent. Therefore, the general 17



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trends of our results can also be compared to the lock and key colloidal experiments, where the depleting agent occupied volume fraction can be tuned by changing its concentration and/or its gyration radius. Experimentally, this can be done by increasing/decreasing its molecular weight or by swelling/shrinking the polymer with temperature (thermosensitive microgels). 5

Figure 6: Fraction of assembled lock-key pairs as a function of the occupied volume fraction as obtained from equation 4 and considering a total lock and key concentration of 0.001M . All data correspond to σk /σs = 4 and charged lock and key particles with zl = zk = 8. Black circles and red squares correspond to a 1:1-electrolyte solution with a concentration of 0.15 M and 0.8 M, respectively. Cyan diamonds correspond to 0.15 M of a 2:2-electrolyte solution. The inset shows the corresponding bonding (contact) energy. The fraction of bonded lock and key pairs, [LK]/[L]0 , can be obtained from the following expression 5 [L]0 [LK] = exp {−β[Ec + ln(vc [L]0 )/β]} [L][K]

(4)

where [LK], [L], and [K] are the concentrations of bonded lock and key pairs, free lock, and free key, respectively. In this expression, the total lock concentration is given by [L]0 = [LK] + [L], the contact energy is Ec , and the lock cavity volume is represented by vc . Expression 4 is yield by considering an ideal behavior for lock, key, and bonded lock-key pairs 18


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(all considered diluted gases), and by accounting for a configurational entropy associated with fluctuations of the relative lock and key position within the cavity. Furthermore, the expression assumes the direct lock key bonding as the only possible associations and neglects the rotational degrees of freedom of the bonded lock and key dimmer. Note also that, in our case, Ec is obtained for the infinite dilute limit, which imposes an additional limitation. Hence, deviations from the experimental system should increase for increasing [L]0 . Fig. 6 is obtained form equation 4 for charged systems with valences zl = zk = 8, a relative size of macro-molecules to the solvent of σk /σs = 4, and restricting the equation to the case [K] = [L]. The cases correspond to different electrolyte solutions. These are 0.15M and 0.80M 1-1 electrolyte solutions, and a 0.15M 2-2 electrolyte solution. The inset show the energy value from which the curves are obtained and a linear fit. As the figure shows, there is practically no effect from increasing the 1-1 electrolyte concentration from 0.15M to 0.8M. Thus, the screening is well achieved with both relatively high concentrations. A better screening is obtained with a 2-2 electrolyte, where a clear shift of the bonded lock-key fraction curve to the left is obtained. This shift is a consequence of a much better screening of the repulsive electrostatic force and a relatively small, but clearly significant, decrease of the contact energy Ec for all depleting agent concentrations. Hence, fewer ions are needed to screen the lock-key naked charges and this affect both electrostatics and contact contributions from counter-ions and depletant. Nonetheless, the decreasing rate of Ec , as a function of the volume fraction, ϕ, is almost the same for all systems, as pointed out by the similar slopes shown in the inset, in consistency with the similar qualitative behavior for all the [LK]/[L]0 curves. Hence, electrostatics screening and depletant concentration favor lock-key pairing. In colloidal science it is well known that divalent counterions promotes flocculation, 62–64 even at relatively low concentrations. Our 2-2 electrolyte case here is not an exception. Fig. 7 is obtained similarly to Fig. 6 but for uncharged systems. Again, in this case [K] = [L] is set and σk /σs = 4. No electrolyte is added. The studied cases correspond to a perfect match, σk /σc = 1, a small key σk /σc = 0.7, a large key σk /σc = 1.3, and when no

19




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previously considered charged system. Finally, we emphasize that, in general, Figs. 6 and 7 reproduce the experimentally obtained trends reported by Sacanna et al. 5 for controlling the fraction of lock and key complexation.

4. CONCLUSIONS The depletant forces vs. entropy approaches to self-assembly are complementary: While a force balance is a useful description of local forces, self-assembly responds to the system needs to uniformly distribute its energy. Hence, self-assembly is an entropy driven phenomena. Results were presented in three subsections. In the first one we study a charged lock and key pair immersed in a solvent plus a primitive model electrolyte. We have shown the different force contributions to the directional binding, and how they are influenced by the co-ion and solvent release from the cavity. We have also focused on the energy binding path for macro-particles with different valence. The overall result is that only a large and repulsive direct Coulomb interaction can compensate the binding entropy gain. This situation may occur in highly charged colloidal systems. In the second subsection we have shown under which circumstances the scaling of the macro-particles to the depletant (solvent) size ratio leads to a universal behavior of the binding energy as a function of distance, when properly re-scaled. The universal behavior is obtained only for key sizes larger or equal to the matching lock cavity. Furthermore, the obtained master curves also depend on the particular geometry. Thus, in general, one cannot expect universality to occur. Finally, the third subsection is devoted to qualitatively compare our simulation predictions with the experimental data given by Sacanna et al. 5 The comparison includes the effects of changing the electrolyte concentration, valence, and cavity-key size ratio. In general, a good agreement is found. Notwithstanding, our prediction of a lock and key relatively large binding for a key size below the cavity one, does not agree with the experimental evidence.

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This suggests other effects, such as polymer softness, to play an important role in this case.

Acknowledgments Authors thank CONACyT financial support trough Project No. 169125.

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Entropy Driven Self-Assembly in Charged LockâKey Particles

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