Article pubs.acs.org/Langmuir
Assembly of Lock-and-Key Colloids Mediated by Polymeric Depletant Hung-Yu Chang,† Chang-Wei Huang,† Yen-Fu Chen,† Shyh-Yun Chen,† Yu-Jane Sheng,*,† and Heng-Kwong Tsao*,‡ †
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C. Department of Chemical and Materials Engineering, Department of Physics, National Central University, Jhongli, Taiwan 320, R.O.C.
‡
S Supporting Information *
ABSTRACT: Polymer-mediated lock-and-key assembly via depletion attraction is purely a shape-recognition process without any molecular bonding. Since the depletion attraction relates to osmotic pressure and excluded volume, the binding tendency in a dispersion of lock-and-key colloids can be controlled by adjusting the characteristics of polymeric depletants. In this work, dissipative particle dynamics accounting for explicit solvents, polymers, and multiple lock−key pairs are performed to investigate the influences of the polymer concentration, chain length, solvent quality, and chain stiffness. As the polymer concentration associated with osmotic pressure is increased, the binding free energy (Eb) between a lock−key pair rises linearly and the binding fraction (θLK) in the dispersion grows sigmoidally. Moreover, the increases in the chain length, solvent quality, and chain stiffness lead to the expansion of the polymer size associated with excluded volume and thus both Eb and θLK rise accordingly. However, Eb and θLK grow to be insensitive to the chain length for long enough polymer coils but still can be enhanced if the polymer becomes rod-like. As the solvent quality is varied, θLK can be dramatically altered, although the radius of gyration of polymers is slightly changed.
I. INTRODUCTION A stable colloidal dispersion can be driven to aggregate by the addition of nonadsorbing polymers.1−3 This polymer-mediated attraction is the so-called depletion force, suggested first by Asakura and Oosawa.4,5 The physical origin of the depletion force is associated with the overlap of the excluded volumes of the two colloidal particles, which leads to the increase of the system volume accessible to polymeric depletants and solvents. As a result of the aggregation of colloidal particles, the entropy of depletants grows and thus the system free energy declines. Evidently, such an attraction is entropically driven and it depends only on the size, shape, or surface roughness of colloids, not on their chemical composition and surface functionalities.6−10 By means of the entropic depletion, aggregation of colloids can be induced and thereby the separation of colloids can be achieved. For example, the depletion attraction induced by polymers has been used in the shape- or size-dependent separation of nanoparticles recently,11−13 in addition to traditional biological applications such as protein extraction and purification. In general, the depletion attraction between colloidal particles lacks selectivity and directionality. In contrast, the interactions between receptors and ligands such as enzymes and substrates are highly specific and are often described by the lock-and-key model.14,15 The lock−key (LK) self-assembly formed in the association process, L + K ⇌ LK, is a © XXXX American Chemical Society
characteristic in numerous biological processes, and the binding is driven by enthalpic interactions such as hydrogen bonding, dipolar, and electrostatic attractions.16−18 According to Fischer’s lock-and-key principle,14 the geometrical complementarity between the cavity of an enzyme (the lock) and a substrate (the key) guarantees the specific biochemical reactions. In fact, such a LK shape recognition between antigens and antibodies plays a crucial role in triggering the therapeutic immune responses.19 Recently, a series of LK assembling structures driven by the depletion force have been demonstrated in experiments,20,21 including multiple locks around a single key, colloidal wormlike chains via lock−lock association, and snowman-like particle assemblies. When nonadsorbing water-soluble polymer (depletant) is added to the system, the charge stabilized lock-and-key colloids tend to bind with each other due to the short-range depletion attraction. The LK assembly involving the spherical key fitting inside the lock cavity is similar to a ball-and-socket joint held together by the depletion attraction. The absence of chemical bonds in the joint allows the key to rotate within the cavity of the lock, and this rotation is observable under an optical microscope.20 Benefits from depletion interactions Received: July 9, 2015 Revised: November 12, 2015
A
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repulsion and rc the cutoff radius. r̂ij is the unit vector in the direction of the separation and rij the distance between the two beads. vij is the velocity vector of bead i relative to bead j. γ depicts the friction coefficient, and σ = (2γkBT)1/2 is the noise amplitude. ξij denotes a random number whose mean is zero. ωD and ωR are rij-dependent weight functions, and typically one adopts ωD = (ωR)2 = (1 − rij/rc)2. Those forces are soft and pairwise-additive and have the same interaction range rc. Except solvent beads, polymeric chains and lock-and-key colloids are bound together by harmonic spring forces with the spring constant 100 and the equilibrium length 0.7. Note that all the units are scaled by the bead mass m, cutoff distance rc, and thermal energy kBT. Lock-and-key colloids with complementary shapes are constructed by a cluster of DPD beads arranged in a body-centered cubic structure. As shown in Figure 1a, the
between lock-and-key colloids are not only the shape complementarity but also the rotational freedom, which allows for the future application of the microscale machinery with a mobile joint.20,22 This shape-directed assembly methodology could also be employed to separate a desired set of particles from a colloidal mixture without external stimuli or chemical reactivity. In comparison with conventional ultracentrifugation and size-exclusion chromatography,23,24 this assembly process based on depletion attraction and shape complementarity is relatively simple, inexpensive, and more amenable to mass production of materials. In order to grasp microscopic insight into the mechanisms of the entropic-driven LK assembly processes, a number of theoretical and computational studies are also performed.25−35 Typically, these models consist of a LK pair immersed in small hard spheres (depletants) and the depletion interaction is calculated by hypernetted-chain (HNC) integral equation theory,25−27 density functional theory (DFT),29 and Monte Carlo (MC) simulations.30,32 For example, the HNC equations have been employed to evaluate the spatial distribution of the depletion potential between a hard cube having a hemispherical cavity (lock) and a sphere (key).25−27 Moreover, a hybrid MCDFT method has been proposed to quantify multidimensional entropic forces in complex molecular systems.29 The entropydriven attraction associated with the depletion can be simply modeled as the binding (free) energy Eb. It can be acquired from a LK pair and often used to estimate the equilibrium binding constant (Keq), which is related to the concentrations of free key [K], free lock [L], and bound lock−key [LK] by20,32 Keq =
[LK] = Vbe E b / kBT [K][L]
(1)
Here Vb is the binding volume and related to the cavity size. The direct calculation of Vb is sophisticated, and thus, the binding constant is generally determined from a dispersion of lock-and-key colloids. Although the kinetics about direct and indirect routes to the binding has been investigated recently,36 the theoretical study of assemblies of a lock-and-key colloidal dispersion is still limited. In this work, dissipative particles dynamics is exploited to investigate the nanodispersion of lock-and-key colloids that bind spontaneously in the presence of nonadsorbing polymeric depletants. The influences of the polymer characteristics on the assembly behavior of lock-and-key colloids are thoroughly examined. The effective depletion interaction is evaluated via a direct calculation of the force at a specified interparticle distance in a system consisting of a pair of lock-and-key colloids only. The binding fractions, θLK = [LK]/([LK] + [L]), are calculated from the equilibrium systems of dispersed lock-andkey colloids. The effects of polymer concentration, polymer length, solvent quality, and chain stiffness on the depletion force, binding energy, and binding fraction are studied.
Figure 1. (a) Schematic diagrams of key colloid, lock colloid, polymeric depletant, and solvent. (b) A snapshot of an equilibrium system of five lock−key pairs.
radius of the key colloid is RK = 2.8 and that of the lock is RL = 4.4. A hemispherical cavity with the radius RC is created on the surface of a lock colloid by removing the DPD beads within the cavity. RC is either 2.8 or 3.5. In order to reinforce the stiffness of the lock-and-key colloids, additional spring forces are also imposed for nonadjacent beads.3,40 To prevent self-clustering among lock colloids, polymeric chains with the length Mg = 3 are grafted on all surfaces of lock colloids. Note that the nonspecific binding36 cannot be observed in this work even for lock colloids without the steric layer because of repulsive forces between DPD beads. Typically, 100 pairs of lock-and-key colloids are employed for the calculation of binding fraction (θLK). For the purpose of demonstration only, a snapshot of an equilibrium system for five lock−key pairs is shown here in Figure 1b. In this work, there are four kinds of DPD beads: solvent (S), nonadsorbing polymeric depletant (P), and solvophilic lock (L)
II. MODEL AND SIMULATION METHOD A coarse-grained particle based simulation, dissipative particle dynamics (DPD), has been employed in this study.37−39 A DPD bead represents a cluster of atoms or molecules and has mass m and diameter rc. All DPD beads obey Newton’s equation of motion and are acted on by conservative (FCij ), dissipative (FDij ), and random forces (FRij ). The forces exerted on bead i by bead j are given by FCij = aij(rc − rij)r̂ij, FDij = −γωD(r̂ij· vij)r̂ij, and FRij = −σωRξijr̂ij. Here aij represents the maximum B
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III. RESULTS AND DISCUSSION For a dilute dispersion, the depletion potential (UD) between two colloidal particles is typically expressed as the product of osmotic pressure (Π) and overlapping exclusion volume (Vex) associated with depletants, UD = ΠVex.20,42 According to the van’t Hoff equation, one has UD = cPkBTVex, where cP is the number density of depletant and kBT the thermal energy. Moreover, the excluded volume is related to the sizes of colloids and depletant. As a result, it is anticipated that for systems with polymeric depletants the depletion attraction grows with the polymer concentration and varies with the cavity size and polymer size which depends on chain length, solvent quality, and stiffness. In this work, the self-assembly behavior of a nanodispersion of key (RK = 2.8) and lock (RL = 4.4) driven by depletion attractions has been investigated by DPD for two different sizes of cavities (RC = 2.8 and 3.5). In the presence of nonadsorbing polymers, the lock-and-key binding is specific and highly relies on the characteristics of the polymer, such as polymer concentration (φP), polymer length (MP), solvent quality (aSP), and chain stiffness. The effective depletion interaction of a pair of lock-and-key colloids only is determined. In addition, the equilibrium states of the lock-andkey colloidal dispersion are achieved and the binding fraction (θLK) is therefore acquired quantitatively. A. Effect of Cavity Size (RC). The depletion force (FD) can be directly evaluated from DPD. The inset of Figure 2 shows
and key (K) colloids. In general, the repulsive interaction parameters (aij) between the same species are set as aSS = aPP = aLL = aKK = 25. To eliminate all but depletion forces, we have simply chosen aij = 25 as i ≠ j. More precisely, all the molecular interactions but steric repulsions are essentially turned off. The steric interaction is described by the short-range repulsion (aij) between two DPD beads. Basically, all colloids can be regarded as hard particles. Moreover, aij is chosen as 25 to ensure that the solvent quality is good for both colloid and depletant. That is, aggregation among colloids or depletants will not occur. Only in the study of the effect of the solvent quality, aSP is changed from 19 to 27. The dynamics of 648 000 DPD beads (Ntotal) is performed in a cubic box (603) under periodic boundary conditions at the system density ρ = 3. The DPD time step is set at Δt = 0.04, and the total DPD steps are at least 2 × 106. The volume fractions of key colloids (φK = NKMK/ Ntotal) and lock colloids (φL = NLML/Ntotal) are about 0.04 and 0.16, respectively. The concentration of polymeric depletant is varied, i.e., φP = NPMP/Ntotal = 0−0.35. Here NK, NL, and NP denote the total number of key colloids, lock colloids, and polymers, respectively. MK, ML, and MP are the numbers of DPD beads for the construction of one key, lock, and polymer. The volume fraction of solvent beads varies from 0.45 to 0.8. In order to quantify the concentration [LK], the bound LK is formed if the center-to-center distance between lock and key is less than RL − RC + RK + δ, where δ = 0.2 typically. The profiles of the depletion force FD(r) and depletion potential energy UD(r) between a pair of lock-and-key colloids can be obtained from the concept of the potential of mean force. That is, the mean depletion force is evaluated from the ensemble average of the force acting on key or lock colloid, which are fixed in space and separated by the interparticle distance r passing through the cavity center. By keeping a set of constituent DPD beads of lock-and-key colloids at given positions {x1, ..., xM}, the force is evaluated over all the configurations of all the remaining DPD beads (solvent and polymer) in the system with {xM+1, ..., xN}. The definition of FD(r) is given by41 FD(r ) =
∫ F(x1, ..., xM)e−βU(x1,..., xN)dxM + 1, ..., dxN ∫ e−βU(x1,..., xN)dxM + 1, ..., dxN
The total force acting on the key colloid is essentially the same as that on the lock colloid, but they are in opposite directions. The separation-dependent depletion force is then acquired by varying the distance between lock and key. As r is large enough, the mean force FD(r) vanishes. The potential of mean force is the average work required to bring a pair of lock and key from infinity to the distance r, corresponding to the interaction free energy between lock-and-key colloids. It can be calculated by the integration of the mean force with respect to their separation3,41
Figure 2. Depletion potential energy (UD) and depletion force (FD) plotted against the interparticle distance (r) between a LK pair for RC = 2.8 and 3.5 at MP = 12 and φP = 0.15.
the variation of FD with the interparticle distance (r) between a LK pair for two cavity sizes. Here the polymer concentration φP = 0.15, chain length MP = 12, and solvent quality aSP = 25 are chosen. Only the “convex-to-concave” configuration is considered. The key colloid with RK = 2.8 fits into the lock colloid with the cavity RC = 2.8, and this LK pair is perfectly matched. In contrast, the LK pair with the cavity RC = 3.5 is size-mismatched. The qualitative features of FD of a LK pair with RC = 2.8 are analogous to those with RC = 3.5. As the key is approaching the lock, repulsions and attractions exist. Since the distance associated with the perfect matching between the LK pair is at RL − RC + RK, the region of the attractive force is in the neighborhood of r = RL − RC + RK. Significant hard-core repulsion arises monotonically as the lock-and-key colloids move closer. On the contrary, repulsive−attractive oscillation is revealed as they move further away. Such an oscillatory feature
r
UD(r ) = −
∫∞ FD(r) dr
The binding energy (Eb) refers to the minimum value of UD(r). Note that the entropy of both depletants and solvents is taken into account in the calculation of the potential of mean force. The translational entropy of lock-and-key colloids is included in the equilibrium binding simulation and shown through their concentrations in eq 1 for Keq. C
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the solution of smaller φP owing to the lower binding energy and higher energy barrier. Since the depletion potential is related to the osmotic pressure which in turn depends on the depletant concentration, the binding energy between a LK pair is anticipated to vary with the polymer concentration. Figure 4 shows the plot of Eb
is generally observed for the depletion force between nanoparticles and attributed to local monomer packing correlations.3,43 FD vanishes eventually as the distance is greater than RK + RL. The depletion potential energy UD(r) can be acquired from the integration of FD(r). Figure 2 demonstrates similar features between UD and FD. The primary minimum of UD corresponds to the depletion attraction, and it is referred to as the binding energy (Eb = −UD,max) for the “convex-to-concave” LK configuration. The magnitude of the binding energy with RC = 2.8 (Eb = 28.5) is substantially greater than that with RC = 3.5 (Eb = 18.6). Note that Eb is scaled by kBT. By contrast, the primary maximum of UD acts as a free-energy barrier and it originates from local monomer packing correlations. It is worth noting that the energy barrier with RC = 3.5 (UD = 15.2) is larger than that with RC = 2.8 (UD = 10.6). As a consequence, it is anticipated that the binding tendency of the perfectly matched LK pair is stronger than that of the size-mismatched LK pair because of the higher binding energy and lower energy barrier. B. Concentration Effect of Polymeric Depletants (φP). The inset of Figure 3 shows the plot of FD against the
Figure 4. Variation of the binding energy (Eb) between a LK pair with the polymer concentration (φP) for RC = 2.8 and 3.5 at MP = 12.
against φP for two cavity sizes (RC = 2.8 and 3.5). It is found that Eb is approximately proportional to φP for a given RC. Moreover, Eb associated with the perfectly matched LK pair (RK = RC) is significantly greater than that associated with the size-mismatched one (RK < RC). Our simulation results of the influences of the depletant concentration and the size commensurability between the key and cavity on the binding energy are qualitatively consistent with previous studies.20,32 It is worth mentioning that even in the absence of polymeric depletant the binding energy for the size-mismatched LK pair is as large as about 12.5. This consequence seems to imply that LK binding can be frequently observed for sufficiently high concentrations of colloids. In order to study the depletion-induced lock-and-key association, a dispersion of 100 LK pairs is considered and the equilibrium state is analyzed. For a given polymer concentration, two initial configurations, (1) 100 bound pairs and (2) 100 unbound pairs, are employed in our simulations. At equilibrium, the final binding pairs determined from the two initial configurations are essentially the same and this result ensures that a true equilibrium state is reached. As demonstrated in Figure 5, the variation of binding fraction (θLK) with φP is acquired for two cavity sizes. As expected, the binding fraction grows with increasing polymer concentration. Similar to adsorption, a sigmoidal curve is exhibited. The lockand-key colloids are dispersed (θLK → 0) at sufficiently low φP, but they prefer binding (θLK → 1) at sufficiently high φP. Note that solvent molecules only (φP = 0) can provide depletion attraction as well,44,45 but it is too weak to yield the LK binding in this work. Moreover, at a given polymer concentration, θLK for perfectly matched pairs is larger than that for sizemismatched pairs. These results of θLK agree qualitatively with those of Eb for both polymer concentration and size commensurability. Note that the spontaneous events of lockand-key binding can be seen in the movie of the Supporting
Figure 3. Depletion potential energy (UD) and depletion force (FD) plotted against the interparticle distance (r) between a perfectly matched LK pair for φP = 0.05 and 0.35 at MP = 12.
interparticle distance (r) between a LK pair at φP = 0.05 and 0.35 for MP = 12 and aSP = 25. The sizes of key and cavity are matched RK = RC = 2.8. The qualitative features of a LK pair at φP = 0.05 are similar to those at φP = 0.35. The attractive force occurs at 4.3 ≤ r ≤ 4.7, hard-core repulsion grows monotonically as r ≤ 4.3, and oscillatory repulsion−attraction is seen for 4.7 ≤ r ≤ 5.8. FD diminishes gradually for r ≥ 5.8. Lowering the polymer concentration essentially leads to a small upward shift in the force curve. The major difference between φP = 0.05 and 0.35 occurs at the maximum attraction, FD = −149 and −177. The profile of the depletion potential energy is depicted in Figure 3. The binding energy (energy well) at φP = 0.35 (Eb = 34.8) is significantly higher than that at φP = 0.05 (Eb = 23.8). In contrast, the energy barrier at φP = 0.05 (UD = 12.8) is larger than that at φP = 0.35 (UD = 7.5). As a result, it is easier for the lock-and-key colloids to bind together at higher polymer concentration, whereas they tend to be dispersed in D
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Figure 6. Depletion potential energy (UD) and depletion force (FD) between a mismatched LK pair (RC = 3.5) plotted against the interparticle distance (r) for MP = 1 and 12 at φP = 0.20.
Figure 5. Variation of the binding fraction (θLK) with the polymer concentration (φP) for RC = 2.8 and 3.5 at MP = 12. The equilibrium snapshots of bound LK pairs with RC = 3.5 at φP = 0.15 and 0.30. Free lock-and-key colloids and the depletants are not shown for clarity.
longer chain length is higher than that with a shorter one owing to the higher binding energy and lower energy barrier. The influence of the depletant length on the binding energy is further examined by varying MP systematically. The plot of Eb against MP is depicted in Figure 7 for the size-mismatched LK
Information. A system of only five lock−key pairs is shown for demonstration purposes. By analyzing the concentration of the bound LK pair at equilibrium, one can acquire the equilibrium binding constant. On the other hand, Keq can also be inferred from eq 1 if the binding energy and binding volume are obtained. Unfortunately, the binding volume which represents the orientational freedom and translational volume of the bound LK is difficult to estimate directly.18,46 For RK = RC = 2.8 at φP = 0.05, one has Keq ≈ 1.5 × 10−3 associated with θLK = 0.12 revealed in Figure 5 and Eb = 23.8 as shown in Figure 4. This small binding fraction is not consistent with the large binding energy. In fact, the binding fraction is close to zero but the binding energy is as large as about 22 for φP = 0. Note that the volume fraction of lock-and-key colloids is not low (φK + φL = 0.2). According to eq 1, the binding volume is then calculated, Vb ≈ 6.9 × 10−14. Such a small value of Vb corresponds to a huge loss of rotational and translational entropy associated with the formation of the lock-and-key assembly. When the binding volume is small, it impedes the lock-and-key association strongly and the strong binding energy acquired from the potential of mean force of a LK pair is not enough. As a result, the effective binding energy should be considered as Eb + ln(Vb). C. Effect of Depletant Length (MP). It is known that the depletion potential is proportional to the excluded volume Vex which is related to the depletant size. For polymeric depletant, Vex is dependent on its radius of gyration and thereby UD is affected by the chain length. The inset of Figure 6 illustrates the curves of FD(r) between a size-mismatched LK pair (RC = 3.5) for MP = 1 and 12 at φP = 0.2. Here aSP = 25 is adopted, and the chain length (MP) is represented by the number of DPD beads constructing the polymer. The increment of the chain length results in a small downward shift in FD. The main difference between MP = 1 and 12 takes place at the maximum attraction, FD = −148 and −170. The distribution of the depletion potential energy is demonstrated in Figure 6. At the same φP, the binding energy (energy well) for MP = 12 (Eb = 21.1) is substantially greater than that for MP = 1 (Eb = 13.1). On the contrary, the energy barrier for MP = 1 (UD = 18.4) is larger than that for MP = 12 (UD = 12.3). The binding tendency of the lock-and-key colloids in the polymer solution with a
Figure 7. Variation of the binding energy (Eb) between a mismatched LK pair with the chain length (MP) at φP = 0.20 and 0.25. The variation of the radius of gyration (Rg) with MP is shown in the inset.
pair at φP = 0.20 and 0.25. At the same volume fraction of polymers, the binding energy grows rapidly for shorter chains (1 ≤ MP ≤ 6) but seems to approach a plateau for longer chains (MP > 6). As a result, the depletant concentration associated with the osmotic pressure seems to be more influential than the depletant size associated with the excluded volume in determining the lock-and-key assembly for long enough polymers. The effect of MP on θLK is also illustrated in Figure 8. For small depletant (MP = 1), θLK is essentially zero and lockand-key colloids are well dispersed in the system. As MP is increased at a given φP, θLK grows accordingly. However, for long enough polymers, the variation of θLK with φP which exhibits a sigmoidal curve is essentially the same for 9 ≤ MP ≤ 18, as demonstrated in the inset of Figure 8. Note that the radius of gyration changes from 0.5 to 1.5 as the chain length grows from MP = 3 to 18, all of which are smaller than the E
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LK pair with MP = 12 and φP = 0.15. Evidently, the binding energy for aSP = 19 (Eb = 24.0) is greater than that for aSP = 27 (Eb = 16.2). On the other hand, the energy barrier for aSP = 19 (UD = 12.9) is smaller than that for aSP = 27 (UD = 17.5). These results reveal that a better solvent gives the higher binding energy and lower energy barrier. The variation of Eb with aSP is displayed in Figure 9. Note that aSK and aSL remain to be 25. It is found that, when aSP is increased, the binding energy descends monotonically. As the solvent quality is getting poorer by increasing aSP, the polymer size shrinks, the excluded volume effect decreases, and the depletion attraction is weakened. Therefore, in poor solvents (larger aSP), lock-andkey associations are anticipated to be inhibited. The equilibrium state of a dispersion of lock-and-key colloids is also studied for different solvent qualities. Figure 10 shows Figure 8. Binding fraction (θLK) for mismatched lock-and-key colloids as a function of the polymer concentration (φP) at different chain length (MP).
cavity size Rc = 3.5. Our simulation outcomes indicate that both Eb and θLK rise fast with increasing MP for shorter polymers but become insensitive to MP for longer polymers. D. Effect of Solvent Quality of Polymers (aSP) and Chain Stiffness. The size of polymeric depletant depends not only on its chain length but also on the solvent quality and polymer stiffness. Generally, the solvent quality of a polymer manifested through its swelling capacity can be tuned by altering the solvent medium. For thermoresponsive polymers such as poly(N-isopropylacrylamide), another feasible way to manipulate their solvent quality is the change of temperature.47 The polymer conformation varies from a random coil to a collapsed globule when temperature exceeds its lower critical solution temperature. By using thermoresponsive polymers as depletants, the self-assembly of lock-and-key colloids can be facilely controlled via switching temperature.20 In our DPD simulations, the solvent quality of the polymeric depletant is manifested by the repulsive interaction parameter between solvent and polymer (aSP).48 That is, the solvent quality rises as the value of aSP declines. The depletion potential energy UD(r) for different solvent qualities is shown in the inset of Figure 9 for a size-mismatched
Figure 10. Binding fraction (θLK) for the mismatched lock-and-key colloids as a function of the solvent quality (aSP) at MP = 12 and φP = 0.15. The radius of gyration of polymer (Rg) plotted against aSP is shown in the inset.
the plot of the binding fraction against aSP, and the variation of the radius of gyration of polymers (Rg) is illustrated in the inset. Both θLK and Rg curves decay monotonically as the solvent quality decreases (corresponding to an increase in aSP). Note that polymers do not aggregate at aSP = 27. This consequence implies that lock-and-key colloids tend to bind when polymeric depletants are in a swollen state but incline to separate as polymers turn into a slightly collapsed situation. It is interesting to know that, while the reduction of Rg corresponding to the change of aSP from 19 to 27 is less than 15%, the binding energy is significantly lowered from 24 to 16. Moreover, the binding fraction varies from θLK = 0.96 to 0.04. This result indicates that the binding fraction is very sensitive to the solvent quality even though the range of aSP studied corresponds to good solvents in which no phase separation is observed. The influences of the chain stiffness on the lock-and-key assembly have also been investigated by considering rod-like depletants. To construct a rigid polymer in our systems, two more spring forces with the same spring constant 100 but distinct equilibrium lengths 2 × 0.7 and 11 × 0.7 are added to reinforce the stiffness in the polymer of MP = 12. The mean end-to-end distance (the distance between the first and last beads) of the rigid depletant is 7.5. The comparison of UD(r) for a size-mismatched LK pair between rigid and soft polymers
Figure 9. Variation of the binding energy (Eb) with aSP for a sizemismatched LK pair at MP = 12 and φP = 0.15. The UD(r) curves for aSP = 19 and 27 are shown in the inset. F
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Langmuir is shown in the inset of Figure 11 for φP = 0.10 and aSP = 25. The stiffness of the chain leads to a downward shift of UD(r) of
The depletion force can be directly acquired from the potential of mean force. As the key is approaching to the lock, there exist repulsions and attractions. Such an oscillatory feature is attributed to local monomer packing correlations. The depletion potential energy is obtained from the integration of the depletion force, and the primary minimum and maximum correspond to the binding energy and energy barrier. The perfectly matched LK pair possesses a higher binding energy and a lower energy barrier than those of the sizemismatched LK pair. As the concentration of polymeric depletant is increased, Eb rises linearly while θLK grows sigmoidally. By analyzing the equilibrium concentrations of free colloids and bound pairs, the equilibrium binding constant in eq 1 is acquired as well. It is found that, at low polymer concentrations, the small θLK associated with the large Eb is due to the extremely small value of the binding volume. That is, the lock-and-key association is strongly hindered because of a great loss of orientational freedom and translational volume associated with the LK assembly. The effective Eb should take into account the contribution of the binding volume. In addition to the depletion concentration associated with the osmotic pressure, the depletion potential is also affected by the depletant size associated with the excluded volume. The characteristic size of the polymeric depletant depends on the chain length, solvent quality, and chain stiffness. As the chain length is increased, both Eb and θLK rise fast for shorter polymers but become insensitive to the chain length for longer ones. The association of lock-and-key colloids is sensitive to the solvent quality. As the solvent quality is decreased, both Eb and θLK decay monotonically, although the reduction of the radius of gyration of polymers is not significant. The influence of the chain stiffness is studied by considering the rod-like polymer. Since Eb in the rod-like polymer solution is higher than that in the coil-like one, θLK of a lock-and-key dispersion can be enhanced by using rod-like depletants.
Figure 11. Binding fraction (θLK) for the mismatched lock-and-key colloids as a function of the polymer concentration (φP) for different chain stiffness at MP = 12. The UD(r) curves at φP = 0.10 for different stiffness are shown in the inset.
coil-like polymers. In other words, the energy well in the rodlike polymer solution (Eb = 18.5) is deeper than that in the coillike one (Eb = 14.4). Contrarily, the energy barrier in the coillike polymer solution (UD = 18.4) is higher than that in the rod-like one (UD = 13.7). This result reveals that increasing the stiffness promotes the association of lock-and-key colloids. As anticipated, Figure 11 shows that the binding fraction of lockand-key colloids in a rod-like polymer solution is always greater than that in a coil-like one. This outcome can be explained by the increase of the effective excluded volume associated with the rod-like depletant. In DPD simulations, since a solvent bead can contain a lot of solvent molecules, the colloids in our simulations can be nanosized or larger. For micron-sized lock-and-key colloids in experiments, nevertheless, we believe that our results can capture the essential features of realistic cases. One can predict the binding constant (Keq) and binding energy (Eb) in realistic cases if the dependence of Keq and Eb on the colloid size is established at a given polymeric depletant concentration. Moreover, the kinetics of the lock-and-key assembly mechanism can be directly studied by DPD simulations, including the rates of formation and breakage of bonds between lock-and-key colloidal particles and the route to bond formation.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b02527. The movie demonstrates the spontaneous binding process of five free key (RK = 3.6) and five free lock (RL = 4.4 and RC = 4.0) colloids in the presence of polymers (MP = 12 and φP = 0.30). Two lock-and-key pairs are held together after 500 000 steps by the depletion attraction. For clarity, polymeric depletants are not shown in the movie. (MPG)
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IV. CONCLUSIONS The association of lock-and-key colloids via the depletion interaction is based on complementary shapes and independent of material composition and chemical reactivity. The strength of the binding can be tuned by adjusting the depletant characteristics. In this work, the influences of the characteristics of the polymeric depletants on the assembly of a dispersion of lock-and-key colloids are explored by DPD simulations accounting for explicit solvents, polymers, and multiple lock− key pairs. On the basis of the direct evaluation of the binding energy (Eb) between a LK pair and the analysis of the binding fraction (θLK) at the equilibrium state of dispersed lock-and-key colloids, the effects of polymer concentration, chain length, solvent quality, and chain stiffness are thoroughly studied.
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected] (Y.-J.S.). *E-mail:
[email protected] (H.-K.T.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research work is supported by Ministry of Science and Technology of Taiwan. Computing times provided by the National Taiwan University Computer and Information Networking Center are gratefully acknowledged. G
DOI: 10.1021/acs.langmuir.5b02527 Langmuir XXXX, XXX, XXX−XXX
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Langmuir
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DOI: 10.1021/acs.langmuir.5b02527 Langmuir XXXX, XXX, XXX−XXX
