Efficient Molecular Approach to Quantifying Solvent-Mediated

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An Efficient Molecular Approach for Quantifying Solvent-mediated Interaction Hongguan Wu, Yu Li, Damir Kadirov, Shuangliang Zhao, Xiaohua Lu, and Honglai Liu Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02629 • Publication Date (Web): 22 Sep 2017 Downloaded from http://pubs.acs.org on September 26, 2017

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An Efficient Molecular Approach for Quantifying Solvent-mediated Interaction Hongguan Wu1, Yu Li1, Damir Kadirov1, Shuangliang Zhao1,*, Xiaohua Lu2, and Honglai Liu3 1

State key laboratory of Chemical Engineering and School of Chemical Engineering, East China University of Science and Technology, Shanghai, 200237, P. R. China 2

State Key Laboratory of Materials-Oriented Chemical Engineering, Nanjing Tech University, Nanjing, 210009, P. R. China

3

School of Chemistry and molecular Engineering, East China University of Science and Technology, Shanghai, 200237, P. R. China

*

Corresponding author: [email protected] 1

ACS Paragon Plus Environment

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Abstract The solvent-mediated interaction, or equivalently the depletion force, play a pivotal role in the processes, by which two objects in solution such as lock and key particles, antibody and antigen, macromolecule and substrate, are attracted to each other. The quantification of this interaction is important yet challenging since it depends on the microscopic solvent structure in the surrounding. Here, we report an efficient molecular approach for predicting the solventmediated interaction by combining the classical density functional theory with a reversible solvation thermodynamic circle. For demonstration, the solvent-mediated interactions between two nanoparticles and between a nanoparticle and a rough wall are examined, and good agreements compared with the simulation results are illustrated. This approach is thereafter employed to interpret the reported self-assembly phenomena of lock and key colloidal particles. We show that the binding probability between the lock and key colloids can be successfully characterized at different depletant concentrations and system temperatures. This approach provides a potential route for identifying the coarse-graining interaction between two objects in fluid systems. Key words: Depletion force; Density functional theory; Mesoscale; Solvation free energy

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I.

Introduction Potential of mean force (PMF), also called solvent mediated interaction or depletion force,

refers to the mean interacting energy between two objects in a solvent [1, 2], and it plays a crucial role in determining the mechanisms of molecular recognition transport

[3]

, binding, protein mediated

[4]

, enzyme catalysis and so on. In addition, the manipulation of PMF also provides

helpful insights to design functional nanomaterial from bottom to up.[5-7] Whereas the concept of PMF was proposed several decades ago

[1]

, recent experiments

[8-11]

and theoretical studies [12-16]

demonstrated the great potential of its novel applications in diverse aspects. Kim et al.[7] stabilized the Pickering emulsions by tailoring the solvent-mediated interaction. They showed that by adding water-soluble polymers as depletant for generating an attractive force between oil droplet and particle, a Pickering emulsion with high internal surface could be prepared. Similarly, by enhancing the PMF between different components in PEO solutions, Wang et al. [9] fabricated a three-dimensional material. Although the manipulation of PMF brings interesting applications, its quantification has been a challenging task. Toward this end, both experimental and theoretical approaches have historically been developed. The first experimental attempt was probably made by Crocker and his coworker [17], and they demonstrated that the PMF in colloidal systems could be measured by using a line-scanned optical tweezer. They argued the force between two colloidal particles became much pronounced with the increased concentration of depletant in solution. Recently, Ma et al. developed a characterization method with atomic-force microscopy.[18] They quantified the hydrophobic interaction, and showed that the replacement of ionic groups on the adlayer of a support could significantly alter the force between a tip and the support.

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Instead of direct experimental measurement, several theoretical approaches have been proposed to evaluate PMF. In general, PMF is associated with the microscopic solvent structure and it can be conventionally defined as [19], W ( r ) = − k BT ln[ g ab ( r )] − Vab ( r ) .

(1)

Here W (r ) is the PMF between the objects a and b in solution, and Vab (r ) is the direct interaction in-between, which can be described with standard force field. g ab (r ) is the radial distribution function (RDF), stating the probability of finding object b in a position given the object a locating at another position with separation r .[20] When the concentrations for the components a and b are finite, the RDF g ab (r ) can be readily collected from standard computer simulations

[13, 15, 21, 22]

, and thereafter the PMF can be determined straightforwardly through

eq.(1). However, when the concentrations for the components a and b are infinitely small, adequate simulation sampling of RDF g ab (r ) becomes inaccessible, and then RDF is usually calculated by using the integral equation theories [20, 23, 24], g ab ( r ) = exp[ − β Vab ( r ) + hab ( r ) − cab ( r ) + Bab ( r )] .

(2)

The above equation refers to the so-called exact “closure” to the integral relation between the direct correlation function cab and total correlation function hab with hab = g ab − 1 . In eq.(2), Bab is the bridge function accounting for multi-body correlation.[20] Within the framework of integral equation theories, g ab (r ) can be numerically solved as long as an approximation for the bridge term is given[20, 25]. Recently, Jin et al

[26]

proposed an interesting method to calculate the PMF

between a lock and key molecule with the help of potential distribution theorem. They combined Monte Carlo simulation for collecting the microscopic configuration and density functional

4


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theory for computing the free energy, and demonstrated that this hybrid method could give accurate prediction of PMF in various circumstances comparing to direct simulations. In this work, a new conceptual approach is proposed to quantify PMF with the aid of solvation free energy thermodynamics and density functional theory (DFT). We demonstrate that this approach is free of simulation thus computationally efficient. In addition, it is versatile and applicable for the systems with either finite or infinitely low concentrations of the target component. With the help of this approach, we take hard-sphere solvent systems as a case study, and examine the PMF between nanoparticles and that between a nanoparticle and a rough wall, calibrated with available simulation data. Afterward, we further explain the reported selfassembly phenomena of lock and key colloidal particles [8]. The remainder of this work is organized as follows. In Sec. II, the theoretical basis for this new approach is introduced, followed by the demonstrations of PMF evaluations in different catalogues as presented in Sec. III. The binding of lock-key particles are examined in Sec. IV. Finally, a brief conclusion is given in Sec. V. II.

Theory and method The main idea behind our approach for calculating PMF is based on a reversible solvation

thermodynamic circle. Namely, the PMF is associated with the required reversible work for bringing one object to the other in a solution. This reversible work, composed of the direct interacting energy Vab (r ) and the PMF W (r ) , can be computed by the grand potential difference of the solution. Equivalently, we have

W (r ) = Ω(r ) − Ω(∞) − Vab (r ) ,

(3)

where Ω(r ) or Ω(∞) correspond to the grand potential of the solvent system in the presence of objects a and b as solutes with separation r or departing infinitely away. On the other hand, the

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solvation free energy of a solute can be calculated by the grand potential difference after and before dissolving the solute [25, 27, 28], namely Fsol =Ω − Ω 0 .

(4)

Here Ω and Ω 0 stand for the grand potentials of solvent with and without the addition of the solute. The combination of eqs.(3) and (4) leads to

W (r) = [Ω(r ) − Ω0 ] − [Ω(∞) − Ω0 ] − Vab (r ) = Fsolab (r ) − Fsolab (∞) − Vab (r )

.

(5)

In the above equation, Fsolab (r ) denotes the solvation free energy for the solute: pair of objects a and b with separation r . When the separation goes infinitely large r → ∞ , the solvation free energy goes to Fsolab (∞) . Indeed, in the limit case the solvation thermodynamics of objects a becomes independent to that of object b, and equivalently Fsolab (∞) can be expressed as the summation of the individual solvation free energies, and this gives

Fsolab (∞) = Fsola + Fsolb ,

(6)

where Fsola and Fsolb represent the solvation free energies of objects a and b, respectively. The substitution of eq.(6) into eq.(5) gives the final equation determining the PMF in this work,

W (r ) = Fsolab (r ) − Fsola − Fsolb − Vab (r ) .

(7)

We point out that, by taking into account the thermodynamic equivalence between the solvation free energy of a solute and its excess chemical potential in the solvent [28], eq.(7) can be rewritten as

W (r ) = µexab (r ) − µexa − µexb − Vab (r ) ,

(8)

where µexi denotes the excess chemical potential of solute i (= ab , a and b ) in solvent. The scheme of PMF calculation in our approach is depicted in Figure 1. 6


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Figure 1. Scheme of PMF calculation in this work: the PMF between two objects in a solution is associated with the solvation free energy (excess chemical potential) difference of three solutes in the solution. The first solute is the object pair with a face-to-face distance h , and the second or third solute refers to the individual objects.

Before proceeding the discussion on solvation free energy calculation, we argue that eq.(7) is formally exact, and independent of model systems and of designated methods for determining the solvation free energy (or equivalently the excess chemical potential). Eq.(7) indicates that the PMF depends not only on the solvent microscopic structure and molecular interaction, but also on the macroscopic thermodynamic property, hence it can be regarded as a kind of mesoscale interaction. Moreover, this approach can be readily extended to evaluate the PMF between two objects in inhomogeneous solvent systems, such as fluid in nano-pores, where the confinement likely brings interesting effect on molecular binding. In that case, the positions of both objects, instead of the separation in-between, should be involved. Finally, we argue that this approach is applicable for the systems with either finite or infinitely low concentrations of the target component. Certainly, when the concentration of the target component is finite, this component can be treated as the constituent of solvent mixture, and otherwise it can be treated solely as solute.

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Whereas the determination of solvation free energy of a solute dissolving in a specific solvent at given temperature and pressure is a traditional challenging task in physical chemistry, significant progress on the theoretical prediction has been made due to the advance of liquid molecular theories in recent decades, among which the classical DFT appears to be a remarkably successful tool.[25, 29] Within the framework of classical DFT, the solvation free energy of a solute can be expressed as a unique functional of the solvent density function nearby the solute, and at thermodynamic equilibrium, the solvent density distribution minimizes the solvation free energy functional, giving rise to the solvation free energy.[25] Along this line, Zhao et al

[28, 30, 31]

developed a three-dimensional molecular DFT for predicting the hydration free energies of halide and alkali ions, and Liu et al

[32]

developed the site DFT and predicted the solvation

structures and free energies of 15 molecular analogs of amino-acid side chains in water. Recently, both methods have been extended into the high-throughput predictions of the hydration free energy for different solutes at different thermodynamic conditions

[33-36]

, and the prediction

accuracy is overall satisfactory compared with parallel simulation study or with experimental measurement. Within the framework of DFT, the solvation free energy of a solute in a solvent at temperature T with bulk density ρ0 can be formulated as [25, 31]

Fsol [ ρ (r )] = k BT ∫ [ ρ (r ) ln

ρ (r ) − ρ (r ) + ρ0 ]dr + ∫ drρ (r )Vext (r ) − µ ex ∫ [ ρ (r ) − ρ0 ]dr ρ0

(9)

+ F ex [ ρ (r )] − F ex ( ρ0 ) Here kB is the Boltzmann constant, and ρ (r ) is the solvent density profile nearby the solute.

Vext (r ) is the external potential to a solvent molecule at position r originating from the solutesolvent interaction. µ ex is the excess chemical potential of solvent molecule, and F ex ( ρ0 ) is the

8


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excess Helmholtz free energy of the bulk solvent system. Both µ ex and F ex ( ρ0 ) can be evaluated generally with the help of suitable equation of state. F ex [ ρ (r )] is the excess chemical potential of the inhomogeneous solvent system, namely the one in the presence of solute. The formulation of F ex [ ρ (r )] is the key task in the development of classical DFT. Whereas the size of system volume should be specified during the calculation, its choice is irrelevant to the solvation free energy as long as the system volume is sufficiently large so that the density profile

ρ (r ) at the edge of the inhomogeneous system recovers to the bulk density ρ0 . Notwithstanding the extensive progress on the accurate predictions of solvation free energy in both simple and molecular solvents, here we consider simply hard-sphere systems for the demonstration of the methodology. Virtually, hard-sphere model provides a primary description for the excluded volume effect, and it thus has been widely adopted into studying the physicochemical properties of colloidal systems [37, 38]. Specifically, for a hard-sphere solute with size D dissolving in a hard-sphere solvent with solvent molecule size σ , the external potential can be calculated analytically provided the solute being placed at the origin of coordinate frame,

∞ r < (σ + D) / 2 Vext (r ) = Vext (r ) =   0 r ≥ (σ + D) / 2 Both µ ex and F ex ( ρ0 ) can be calculated with the Carnaham-Starling equation of state

(10) [39]

, and

the excess Helmholtz free energy F ex [ ρ (r )] can be computed by using the modified fundamental measured theory.[40] Since the derivation and demonstration of MFMT have been extensively discussed elsewhere [25, 32, 41], its lengthy expression is omitted here. III.

Validation with Simulation Results

Case I. PMF between two large solute particles 9


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The solvent mediated interaction between two colloidal particles in solution is closely related to the phenomena of system stabilization, colloidal flocculation, coagulation etc. The colloidal system is modelled here with dilute hard-sphere solutes immersed in hard-sphere solvent with solvent molecular size σ . Two target solutes are singled out and marked as 1 and 2 with diameter D1 and D2 , respectively. Apparently, we have V12 (r ) = 0 . For comparison with available simulation results[42], we set D1 = D2 =5 σ , and the solvent packing fraction

η = πρ 0σ 3 / 6 as 0.229. The solvation free energy calculations are performed on 3D grids, and the grid resolution is set as 0.2 σ . Similar to our previous work [28, 31, 32], both forward and inversed Fourier transforms are applied in order to speed up the numerical calculation of the excess Helmholtz free energy. Each computation of the solvation free energy typically costs tens of minutes on a single CPU process, much more efficient than the corresponding calculation with standard simulation.

Figure 2. Predicted PMF from DFT (red line) between two large hard-sphere solute particles immersed in hard-sphere solvent with packing fraction η =0.229 in comparison with Monte Carlo simulation result (circles). The size ratio between solute and solvent particles is 5.

10


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Figure 2 plots the solvent mediated potential in terms of the face-to-face distance h between two solute particles in the hard-sphere solvent. the red solid line is the theoretical prediction from current approach, and the circles stand for the Monte Carlo simulation results from Dickman et al.[42] As can be seen, a satisfactory agreement can be obtained. When the face-to-face distance h is smaller than 0.5 σ , an attractive interaction can be found between the two large solute particle, and when h is larger than around σ , the interaction fluctuates and decays to zero quickly as the distance increases. According to the AO model [43, 44], the attraction originates from the depletion of the solvent, and furthermore, an energy barrier should be stridden across when one particle approaching to the other. This mechanics can be used to explain phase transition phenomenon.[45] Figure 3 shows the solvent reduced density ρσ 3 around both solutes predicted by DFT. Apparently, a layer-by-layer solvent structure near the solute particles can be observed due to the excluded volume effect.

Figure 3. Solvent reduced density profiles near two solute particles in three representative cases predicted from DFT calculation. From a) to c), the fact-to-fact distances are: 0，0.8 σ ，2.5 σ .

Case II. PMF between a solute particle and a flat wall with cavity Next, we consider the solvent-mediated interaction between a colloidal particle and a rough flat. The rough wall is represented with a flat hard wall embedded with semispherical cavities. 11


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Similarly, both the colloid and solvent molecules are depicted with hard sphere particles. In order to compare with the MC simulation results[26], we assume the distance among the cavities in the wall is large enough so that the correlation between the solvent structures nearby any two cavities can be neglected. The solvents packing fraction is set as 0.367. The diameter is 4σ for the colloid particle (denoted as Dkey ), and 5σ for the cavity (denoted as Dcav ). We particularly check the PMF between the colloid particle and the rough wall along the normal line of the wall passing through the geometrical centers of the cavity. When calculating the PMF, the wall is treated as a “solute” with a finite thickness, and placed at one edge of the 3D grid. Since the periodic boundary condition implicitly applies due to the usage of Fourier transform, the wall thickness should be large enough, albeit its value is irrelevant to the PMF, so that the solvent structures on both sides of the wall do not correlate. In our calculation, the wall thickness is set as 3σ , and in this case the solvent local density nearby the cavity is expected not to be disturbed by the presence of fluid on the other side of the wall. Additionally, we set the length of the cubic grid as 24 σ , which is large enough so that the solvent local density at the middle zone of the 3D grid recovers to the bulk density. Figure 4 shows the calculated PMF in function of the reduced face-to-face distance. The face-to-face distance here refers to the shortest distance from the solute particle surface to the cavity valley bottom. The solid red line represents the DFT result from the present work, the blue dashed line is from the hybrid method[26], and the circles are the MC simulation results.[26] Whereas small deviation can be found among three curves, our method can predict the solventmediated interaction rather well. The small deviation likely results from the usage of 3D discrete calculation grid. Namely, when placing the wall and colloidal particle on a 3D grid with mesh size 0.2σ, the face-to-face distance h varies discretely. If the peak point is jumped over, the PMF

12


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barrier will be under predicted. To improve the prediction accuracy, a grid with smaller mesh size can be used, which certainly demands larger computer memory capacity. At short distance, the interaction is attractive between the substrate and solute particle, while at large distance the potential approaches to zero, and vanishes after one correlation length. Besides, a significant repulsion barrier is observed when the face-to-face distance is around σ .

Figure 4. Predicted PMF (red line) between a large hard-sphere solute particle and a flat hardwall with a cavity. The solvent is represented with hard-sphere with packing fraction η =0.367. The blue dashed line presents the result from the hybrid MC+DFT method, and the circles represent the results from Monte Carlo simulation[26] . The figure insert depicts schematically the relative position between the solute and the rough wall.

Figure 5 depicts the reduced solvent density profiles in the XY plane perpendicular to the flat wall. The reduced density is defined as the ratio of the local density with the bulk density. Three representative face-to-face distances are presented, namely h =0, 0.9 σ and 2.4 σ . The turbulence near the wall and solute indicates the solvent structure. It’s to be noted that a layer-bylayer solvent structure on the right edge on the XY plane is observed. This artificial solvent 13


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density profile, owing to the usage of periodic boundary condition, should not alter the PMF value, since its contribution to solvation free energies of the rough wall and of the wall-solute pair cancels with each other. Following the AO model, the solvent density distribution plays a pivotal role in the strength of solvent-mediated interaction. When the face-to-face distance is beyond one correlation length, the solvent density profiles around the wall and the solute particles become independent, and therefore the solvent-mediated interaction vanishes.

Figure 5. Reduced solvent density profiles in the solute-wall systems predicted from DFT calculation. From a) to c), the face-to-face distances are 0, 0.9 σ and 2.4 σ . IV.

Binding probability between a lock and key particle Binding between lock and key particles in solution plays an important role in diverse

chemical processes ranging from reaction kinetic to molecular recognition, self-assembly, and new material development etc.[46-49]. Recently, Sacanna et al

[8]

demonstrated that a novel

experimental route for preparing chain particles can be developed with the help of a spontaneous binding process. Specifically, they synthesized colloidal particles with multi-cavities embedded in each particle, and these colloidal particles (called lock particles here and below) could bind with silica spheres (called key particles) in a polymer solution and thereafter gradually form linear or star-like chain particles. The added polymers formed stable and spherical-like globules in solution and served as depletant. By adjusting the cavity diameter and the depletant 14


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concentration and size, they showed that the binding ratio of the lock and key molecules could be elaborately manipulated. In their experiment, the diameter of lock molecule presented a mean value of 2.6µm while the semispherical cavity had a diameter approximately 1.6µm. The key molecules presented homogeneous particle size varying from 1.57µm to 2.48µm. While the radius of gyration of the polymer globule is temperature-dependent, its diameter is 0.130µm at 300K. The relative molecular weight for each polymer molecule is 600,000. We show below that the binding ratio in this experiment can be well explained by using the PMF concept. Towards this goal, we propose firstly a simple model system in which the solvent is treated as a continuum medium, and the depletant and key particles are modeled as hard spheres. The lock particle is treated as a large hard sphere with an embedded cavity. To quantitatively characterize the experimental system, we set the size of polymer depletant here as

σ , and the diameter of the lock molecule is Dlock = 20σ with a cavity Dcav =12σ, the diameter of key molecule varies, namely different diameters are considered as Dkey =12σ, 14.3σ and 19σ. By setting σ =0.130µm, the virtual sizes of the involved particles in the experiment system can be readily recovered. The binding process between a lock and key particle can be treated as a reversible reaction. If each lock contains one cavity and it can bind only with one key particle, the binding process can be described as

L+K

LK .

(11)

Here the symbols L (and K) represents the single lock (and key) particle. LK stands for the lockkey complex. According to standard thermodynamics, the reaction reaches equilibrium when the chemical potential of the reactants equals to that of the product, namely, µL + µK = µ LK , and this gives the criterion for calculating the binding ratio. 15


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The chemical potential of each component can be obtained from the free energy derivative. Considering a dilute solution, the free energies of both lock and key molecule are solely associated with their configurations within each volume unit. While for the lock-key complex, the free energy comprises additionally of the binding free energy and the fluctuation free energy. While the fluctuation free energy corresponds to the relative fluctuation on configuration between the lock and key molecules within the complex in the volume of cavity [15], the binding free energy is virtually the contact value of the solvent mediated interaction. After a straightforward derivation, the following law of mass action for the binding process can be obtained [8]

[ L0 ][ LK ] = exp − β E − k T ln(υ [ L ]) , { [ b B c 0 ]} [ L][ K ]

(12)

where [LK], [L] and [K], are, respectively, the mole concentrations of the lock-key complex, lock and key particle at equilibrium condition, and [L0] is the initial concentration of the lock molecule. Eb is the binding free energy, and vc is the cavity volume. The mass balance equation gives [ L0 ] = [ L] + [ LK ] .

(13)

The substitution of eq.(13) into eq.(12), together with the assumption [ L ] = [ K ] , gives rise to

κ = [ L0 ]υc e − β E , 2 (1 − κ ) b

(14)

here κ = [ LK ] / [ L0 ] is the binding ratio. The above equation gives the criterion calculating the equilibrium binding ratio. In the experiment

[8]

, the initial concentration of lock particle, i.e.,

[ L0 ] , is 5-200×108 particles per milliliter, and υc = π Dcav3 / 6 .

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It’s to be noted that the lock particle is not spherically symmetrical due to the embeddedness of a cavity, and in addition, the lock particle is expected to rotate in solution. Therefore, the binding free energy between a pair of lock and key particles should be the one statistically averaged over all possible relative configurations between the lock and key particles. Namely,

Eb = Wc (ω )

ω

(15)

Here the bracket stands for the ensemble average over all relative orientation ω , and Wc is the contact value of PMF, i.e., the value of PMF when both objects are in close contact. Eq.(15) can be rewritten as Eb =

∫ W (ω ) g (ω )dω ∫ g (ω )dω c

c

(16)

c

where gc (ω ) is the conditional pair correlation function, stating the probability of finding both objects in close contact with relative orientation ω . Recalling eq.(1), we have

g c (ω ) = exp[− βWc (ω )] .

(17)

Here we have applied Vab = 0 . To carry the integration, the lock particle is placed on the origin point of coordinate frame with the cavity facing z axis, and the key particle moves on the surface of lock particle (see the inset of Figure 6). In this case, we have ω = (θ , φ ) with (θ , φ ) being the Euler angle of the key particle within the frame. Apparently, both Wc (θ , φ ) and

g c (θ , φ ) are symmetrical along the φ direction, therefore

Eb =

∫

π

0

Wc (θ ) exp[− β Wc (θ )]sin θ dθ

∫

π

0

exp[− β Wc (θ )]sin θ dθ

(18)

Figure 6 plots the contact value of PMF at different depletant concentrations and relative orientations. We note that the contact value is sensitive to the θ angle, and embeddedness of a 17


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cavity significantly increases the attraction energy. When the key particle sits in the cavity, the attraction energy reaches maximum. In addition, this maximum value increases almost linearly with the depletant concentration.

Figure 6. the contact values of PMF: a) at different relative orientations characterized with θ angle, b) and at different depletant concentrations when the key particle sits perfectly in the cavity. The figure insets give the schematic representation of the relative configuration.

Figure 7 presents the calculated binding ratio in terms of depletant concentration. The solid line is from our prediction, and the symbols represent the experimental results extracted from the original work by Sacanna et al.[8] Here the experimental results are normalized by dividing their individual limit values at high depletant concentration. The deviation between the theoretical prediction and experimental observations can be found. This deviation likely results from simple representation of the actual experimental system. Indeed, the solvent is treated as continuum in our model system, and the hydrophobic depletant particles are modelled as non-interacting hardspheres, which neglect the attraction among them[50]. Nevertheless, our theoretical predictions generally agree with the experimental observations, owing to the well capture of the short-range

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interaction. When the depletant concentration (and equivalently the packing fraction) increases, the binding ratio between the lock and key particles increases monotonously. Particularly, when the sizes of the key and cavity particles are perfectly matched ( Dkey = Dcav =12 σ ), the binding between the key and lock molecules can occur at a relatively low depletant concentration. Otherwise, a higher depletant concentration is necessitated.

Figure 7. The binding ratio between lock and key particles versus polymer concentration from our molecular approach (lines) and from the normalized experimental results (symbols).

08 presents the depletant density profile around the lock and key. From a) to c), the size ratios between the key particle to depletant are 12, 14.3 and 19, respectively. The polymer concentration is 0.3 g/L. One does not observe the layer-by-layer solvent structure due to the low packing fraction.

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Figure 8. Predicted reduced density profiles of solvent around lock and key with different keycavity ratios. From a) to c), the diameters of the key particle are 12 σ , 14.3 σ and 19 σ . In all three cases, the cavity diameter is of 12 σ , and the key particle is in close contact with the lock.

Finally, the diameter of the polymer depletant is sensitive to system temperature, and it has been observed that the depletant is swollen at low temperature and shrunken at high temperature. We find the relation between the depletant size and temperature in the experimental work

[8]

can be

described empirically,

σ = −26.11672 ⋅ erf [0.22417(T − 307.81834)] + 104.1366 .

(19)

Here σ is in micron meter. The erf[‧] represents the error function. Eq.(19) provides a starting point for examining the temperature dependence of the binding ratio between the lock and key particles. 0 plots the predicted binding ratio versus system temperature at different depletant concentrations in the case of perfect geometrical match. Generally, the binding ratio decreases as the system temperature increases, coinciding with the experimental observation. The intrinsic nature of this temperature-dependence is that the reduced density of depletant decreases as increasing system temperature. Specifically, we notice that the temperature dependence becomes less and less sensitive as the concentration of polymer depletant in the system becomes larger and larger. This result is consistent with the analysis on Figure 7, indicating that at a high 20


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polymer concentration the binding process cannot be manipulated by simply adjusting the system temperature.

Figure 9. Binding ratio versus system temperature between the lock and key particles at different concentrations of polymer depletant in the case of perfect geometrical match. The figure inset depicts that the depletant is swollen at low temperature and shrunken at high temperature.

V.

Conclusion In this work, a new molecular approach is developed for quantifying the solvent-mediate

interaction (or equivalently the PMF) between two objects in solution. By combining the solvation concept with a reversible thermodynamic circle, we show the PMF can be evaluated through the prediction of solvation free energy, while the latter can be calculated efficiently by using classical density functional theory. For demonstration, the PMFs between two colloidal particles and between a colloidal particle and a rough wall are examined, and good agreements with the available simulation results are illustrated. Generally, the PMF is featured with an attraction part in short distance and a decaying part in large distance, and it vanishes after around one correlation length. An energy barrier can be found when the distance is approximately of one solvent size. With the help of this approach, we further show that the dependence of binding ratio

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on depletant concentration and system temperature, observed in a recent experiment on lock and key system, can be semi-quantitatively characterized. The molecular approach developed in this work is free of simulation yet accurate and efficient, we expect it offers a powerful tool for identifying the mesoscale interaction between two objects in fluid systems.

Acknowledgement This work is supported by National Natural Science Foundation of China (No. 91434110, 91334203), the 111 Project of China (No.B08021) and the Fundamental Research Funds for the Central Universities of China (1516005). SZ acknowledges the support of Fok Ying Tong Education Foundation (151069) and the Innovation Program of Shanghai Municipal Education Commission (15ZZ029).

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Figure 1. Scheme of PMF calculation in this work: the PMF between two objects in a solution is associated with the solvation free energy (excess chemical potential) difference of three solutes in the solution. The first solute is the object pair with a face-to-face distance h, and the second or third solute refers to the individual object. 236x60mm (150 x 150 DPI)


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Figure 2. Predicted PMF from DFT (red line) between two large hard-sphere solute particles immersed in hard-sphere solvent with packing fraction η=0.229 in comparison with Monte Carlo simulation result (circles). The size ratio between solute and solvent particles is 5. 225x179mm (150 x 150 DPI)


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Figure 3. Solvent density profile near the two solute particles predicted from DFT calculation. From a) to c), the fact-to-fact distances are: 0，0.8σ, 2.5σ. 268x77mm (150 x 150 DPI)


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Figure 4. Predicted PMF (red line) between a large hard-sphere solute particle and a flat hard-wall with a cavity. The solvent is represented with hard-sphere with packing fraction η=0.367. The blue dashed line presents the result from the hybrid MC+DFT method, and the circles represent the results from Monte Carlo simulation. The figure insert depicts schematically the relative position between the solute and the rough wall. 385x290mm (150 x 150 DPI)


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Figure 5. Reduced solvent density profiles in the solute-wall systems predicted from DFT calculation. From a) to c), the face-to-face distances are 0, 0.9σ and 2.4σ. 152x45mm (300 x 300 DPI)


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Figure 6. the contact values of PMF: a) at different relative orientations characterized with θ angle, b) and at different depletant concentrations when the key particle sits perfectly in the cavity. The figure insets give the schematic representation of the relative configuration. 355x149mm (150 x 150 DPI)


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Figure 7. The binding ratio between lock and key particles versus polymer concentration from our molecular approach (lines) and from the normalized experimental results (symbols). 152x107mm (300 x 300 DPI)


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Figure 8. Predicted reduced density profiles of solvent around lock and key with different key-cavity ratios. From a) to c), the diameters of the key particle are 12σ, 14.3σ and 19σ. In all three cases, the cavity diameter is of 12σ, and the key particle is in close contact with the lock. 152x45mm (300 x 300 DPI)


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Figure 9. Binding ratio versus system temperature between the lock and key particles at different concentrations of polymer depletant in the case of perfect geometrical match. The figure inset depicts that the depletant is swollen at low temperature and shrunken at high temperature. 364x258mm (150 x 150 DPI)


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graphic abstract 166x88mm (150 x 150 DPI)


Efficient Molecular Approach to Quantifying Solvent-Mediated

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