纳米粒子/共聚物混合体系结构的Monte Carlo模拟

刘观峰 黄建花

(浙江理工大学化学系,杭州310018)

纳米粒子/共聚物混合体系结构的Monte Carlo模拟

刘观峰 黄建花*

(浙江理工大学化学系,杭州310018)

基于简立方格点模型,对纳米粒子/共聚物混合体系进行了动力学Monte Carlo模拟研究.每一共聚物链均由一个A珠子和三个B珠子组成,表示为A1B3.A1B3链的两亲性体现为B-B之间的相互吸引作用,同时憎水性的纳米粒子之间也存在相互吸引.通过适当选取纳米粒子与B珠子之间的吸引作用势,观察到两种结构:纳米粒子/A1B3链的核-壳结构和纳米粒子分散在憎水壳层中的A1B3囊泡结构.还研究了这两种结构的动力学演化过程,模拟结果表明在纳米粒子分散于囊泡壳层的过程中A1B3囊泡起模板作用.

动力学Monte Carlo模拟;共聚物;纳米粒子;囊泡

1 Introduction

Amphiphilic block copolymers in aqueous solution are known to self-assemble into a variety of supermolecular structures such as micelles,vesicles,cylinders,and lamellae.Vesicles are of special interest since they are important model systems for biological cells and show potential applications in many areas such as microreactors,microcapsules,and drug delivery systems.1-5They have been successfully used as templates for preparing inorganic hollow spheres.For instance,silica hollow spheres were obtained by adding silicon alkoxide to an aqueous solution of amphiphiles,such as cationic,catanionic,or mixed surfactant vesicles,followed by hydrolysis and polycondensation,or a fast silicification.6-9Sub-micrometer hollow metallic spheres of Ni-P alloy were produced by the chemical reduction over anionic surfactant vesicle templates.10

Encapsulation of hydrophobic nanoparticles,such as hydrophobic Au and CdSe nanoparticles,into the vesicle shell through hydrophobic interactions has been achieved.11,12Be-sides,amphiphilic copolymers can form micelle and hydrophobic nanoparticles can thus be loaded into the micelle to form a core-shell structure.13It can improve the stability and surface chemistry of the nanoparticle core and access unique physical properties that are not possible for nanomaterial alone.14,15

Drug delivery is mainly based on successful encapsulation of drugs with different solubility parameters.Two hydrophobic model materials,fluorescent Dye Nile Red representing the molecular size regime and fluorescent quantum dots representing the nano size regime,were successfully encapsulated into the hydrophobic shell of poly(butadiene)-b-poly(ethylene oxide)vesicles.And the quantum dots were observed to be centered inside the double layer of the vesicle shell.16The comblike poly(ethylene glycol)(CPEG)-g-cholesterol vesicle and the cross-linked vesicle of CPEG and CPEG-g-cholesterol were found to be able to entrap considerably hydrophobic doxorubicin(a general anti-cancer drug)in the shell,which showed a great potential as a cargo of the hydrophobic drug.17

Computer simulations play important roles in understanding the formation mechanism of block copolymer vesicle.18-22The phase behavior of nanoparticle/block copolymer mixed system as well as the distribution of nanoparticles in the microdomain of lamellar copolymers have been simulated.23-28The nanoparticle volume fraction,size,and the interaction strength between nanoparticle and copolymer were found to strongly affect the phase behavior of the mixed system.However,we learn little from simulations about the loading process of nanoparticles into vesicles.

In the present work,we studied the phase behavior of nanoparticle/copolymer vesicle mixed system using dynamic Monte Carlo simulation.Hydrophobic nanoparticles were added into the solution containing a single copolymer vesicle.By changing the interaction between nanoparticle and copolymer chain,two structures,nanoparticle/copolymer core-shell structure and copolymer vesicle with nanoparticles dispersed in the shell,were observed.Their formation processes were investigated in detail.

2 Model and simulation method

Simulations were carried out in a simulation box with size 40×40×40 buried in the simple cubic(SC)lattice.The unit of length in this paper was one lattice size of the SC lattice.Periodic boundary conditions were used in all the three dimensions.Each nanoparticle consisted of one single bead(n),and a diblock copolymer A1B3chain was composed of one hydrophilic bead(A)and three hydrophobic beads(B).Self-avoiding was considered,that is,each bead occupied one lattice site and every lattice site could not be simultaneously occupied by more than one bead.The void lattice sites were considered as solvents(S).For polymer chains,the bond length between sequentially linked beads ranged from 1 to 3on the SC lattice, which was originally proposed by Carmesin and Kremer29.The bond between successive beads along a chain was taken from 26 allowed bond vectors obtained from symmetry operations on the set of{(1,0,0),(1,1,0),(1,1,1)}.In this bond-fluctuation model,the beads did not correspond to specific atoms in a polymer chain but rather to small groups of atoms,and the bonds did not represent specific covalent bonds between two atoms but the linkages between beads instead.

Pairwise nearest-neighbor and next nearest-neighbor interactions with the same strength were considered.The amphiphilic property of A1B3chain was represented by interaction energy εBB=-1 for B-B pair and εAA=εAB=0 for A-A and A-B pairs.This model had been successfully used to study the self-assembly behavior of block copolymer in solution.30,31The choice of these effective pairwise interactions implies a generally accepted assumption that the hydrophobic interaction should be much stronger than other attractive forces between amphiphile beads.32The hydrophobic property of nanoparticles was represented by an attractive interaction between nanoparticles (εnn=-1.5),thus they precipitated from the solution in the absence of copolymers.The interaction for solvent-solvent pair was set as εSS=0,which served as a background.Nanoparticles had an affinity toward B-block.So an attraction εnBbetween nanoparticle and B bead was introduced to investigate the mixture of A1B3vesicle and naoparticles.Other interactions,including εAS,εBS,εnS,and εnA,were fixed to be zero.Similar interaction model was used in the simulation of thermodynamic behavior of particle/diblock copolymer mixtures.33The system temperature was fixed at kBT=1.25 in which kBis the Boltzmann constant.Only εnBwas variable in the simulation.

The volume fraction of A1B3chains,φp,and that of nanoparticles,φn,were defined as φp=4Np/V and φn=Nn/V,respectively. Here V was the total sites of the simulation box,Npwas the number of A1B3chains,and Nnwas the number of nanoparticles.The dynamic simulation was achieved by randomly choosing one bead and randomly moving to one of its six nearest neighbor sites.This trial move was accepted if the following conditions were satisfied:(1)the self-avoidance was obeyed, which meant that it could only exchange with a vacancy;and (2)the Boltzmann factor exp(-ΔE/kBT)was greater than a random number uniformly distributed in the interval(0,1),where ΔE was the change in energy due to the trial move.The second criterion,i.e.,the Metropolis criterion,ensured that the system obeyed Boltzmann statistics.For A1B3chain,two additional conditions should be satisfied:(1)the new bond vector still belonged to the allowed bond set;and(2)two bonds did not intersect.Each trial move is called a bead cycle,one Monte Carlo step(MCS)consists of(4Np+Nn)bead cycles.

In the present work,we studied the mixed system containing 7%A1B3chains and 3%hydrophobic nanoparticles.Due to the attraction between B-B beads,A1B3chains self-assembled into vesicles at the volume fraction of A1B3in the range of 4%-15%,22as that used in other simulations.30This was one reason why we chose A1B3in this work.This diblock copolymer A1B3mimics poly(styrene)-b-poly(acrylic acid)(PS-b-PAA)used in experiments.3The size of one bead was taken as the Kuhn length of polyacrylic acid,which is about 1.5 nm or approximately the length of 6 monomers.34The effect of the attraction εnBbetween nanoparticle and B bead on the structure of mixed system was studied from the same start situation.Simulations were carried out as follows:A1B3chains were first randomly put into the system and one single vesicle was formed after a long time of movement.Then nanoparticles were randomly added into the A1B3vesicle system.The simulation time was set as t=0 MCS at this moment.In addition,the attraction εnBbetween nanoparticle and B bead was taken into account. We found different kinds of complex structures by varying the attraction value of εnB,and discussed the mechanism for forming different structures.

3 Results and discussion

Fig.1(a)presents A1B3vesicle formed at φp=7%.It is fabricated by two layers of A1B3chains with A bead locating at the inner and outer surfaces to prevent the contact between B bead and solvent.The vesicular core is filled with solvent.The shell thickness is about 5,while hydrophobic nanoparticles aggregate into a compact sphere to avoid solvent contact.Fig.1(b) presents the compact sphere aggregated at φn=3%in the absence of block copolymer.The radius of sphere is about 8.

3.1 Structures of nanoparticle/A1B3chain system

The structures of A1B3vesicle and hydrophobic nanoparticles mixed system were studied by varying εnBvalues.Fig.2 shows the final structures formed at different εnBvalues.We first consider a limit case that the attraction between nanoparticle and B bead is weak.In this case,nanoparticles assemble into a compact aggregate without any contacts with the vesicle. It is formed by fusion of small aggregates as we previously reported.35Fig.2(a)shows the final structure of the system at εnB=-0.5.Because the attraction εnB(-0.5)is weak,it cannot overcome the attractions between nanopaticles and that between B-B beads.ThusA1B3vesicle remains unchanged.

Fig.1 (a)Vesicle formed byA1B3chains at φp=7%with εBB=-1, (b)aggregate formed in pure nanoparticle/solvent system at φn=3%with εnn=-1.5Blue and red beads representAand B beads ofA1B3chain,respectively,and green ones represent nanoparticles.The same symbols are used in the remaining figures.

Fig.2 Five structures formed with different εnBvaluesεnB:(a)-0.5,(b)-0.6,(c)-0.9,(d)-1.2,(e)-1.6

Turning to the other limit case with εnBclose to or beyond εnn=-1.5,we find that nanoparticles fully enter into the shell of A1B3vesicle as shown in Fig.2(e)at εnB=-1.6.This structure is consistent with experimental observation.11,12,16Therefore one may obtain a hollow sphere after removing copolymers.6-8In this case,nanoparticles can either disperse into the vesicle shell or aggregate into a sphere in the view of energy.However,dispersion into the shell gets more contacts between nanoparticle and B bead,thus leading to the decrease of the system energy.

Three other kinds of structures are formed at moderate εnB=-0.6,-0.9,-1.2.At εnB=-0.6,though nanoparticles still assemble into a compact aggregate,there are a number of contacts between nanoparticle and B bead.Such contacts break the vesicle as shown in Fig.2(b).When εnBis close to εBB,B beads like to contact with nanoparticles as well as with themselves. Since εnnis more negative than both εnBand εBB,nanoparticles still assemble into a compact aggregate.In this case,the nanoparticle aggregate is fully enveloped by A1B3chains, which prevents the contact between nanoparticle and solvent and lowers the system energy.A perfect core-shell structure is presented in Fig.2(c)at εnB=-0.9.With a further increase of the attraction between nanoparticle and B bead to the case|εBB|＜|εnB|＜|εnn|,B bead favors to contact with nanoparticle,but nanoparticles still favor to aggregate with themselves.Thus it is observed that nanoparticles are not well dispersed but aggregate together irregularly in the vesicle shell,as shown in Fig.2 (d)with εnB=-1.2.The irregular aggregate has a large surface to balance the n-B interaction and n-n interaction.In all these three structures,nanoparticles always aggregate together with themselves since the attraction strength|εnB|is smaller than|εnn|.

It is clear that the final structures are dependent on the interactions that we take into account in the system.These structures are controlled by the competition among three attraction strengths εBB,εnn,and εnB.In the present model with εBB=-1 and εnn=-1.5,we observe five structures at different εnBvalues.At small εnBclose to 0,we observe separate nanoparticle aggregate and A1B3vesicle.With the increase of|εnB|,nanoparticle aggre-gate and A1B3irregular aggregate,nanoparticle/A1B3core-shell structure,and A1B3vesicle with nanoparticles aggregate in the shell are observed.And at εnBclose to-2,we observe A1B3vesicle with nanoparticles well-dispersed in the shell.We have also studied the influence of the nanoparticle volume fraction φnon the structure of nanoparticle/A1B3vesicle system.For the case φp=7%,all the five structures can be formed at φn＜10%, and the boundaries between different structures are roughly independent of φn.

Among these five structures,we find that the core-shell structure and nanoparticle-dispersed vesicle are of most interesting.These two structures are widely observed and discussed in experiments,since they are of great potential to carry drug, magnetic and optical particles.11-17It will increase the compatibility and stabilization of nanoparticles after being loaded into copolymer superstructures.In the present simulation,the coreshell structure is obtained by adding nanoparticles into a vesicle system,which is different from experiment where nanoparticles were added into a micellar system.13The evolutions of core-shell structure and nanoparticle-dispersed vesicle are studied in the following.

3.2 Evolution of nanoparticle/A1B3complex structures

We first investigated the evolution of core-shell structure at εnB=-0.9 shown in Fig.2(c).Fig.3 presents the snapshots captured at different periods.Due to the attraction between nanoparticles,they quickly assemble into small aggregates upon addition into A1B3vesicle system.Because of the attraction between nanoparticle and B bead,some nanoparticle aggregates contact with the vesicle,as shown in Fig.3(a)captured at t=0.1×106MCS.Then nanoparticle aggregates become larger with the time going,while the vesicle becomes smaller because more and more A1B3chains are adsorbed onto nanoparticle aggregates(see Fig.3(b-e)).At t=8×106MCS,A1B3vesicle disappears and all A1B3chains envelop around one big nanoparticle aggregate,forming a perfect nanoparticle/A1B3chain core-shell structure,Fig.3(f)shows the final structure formed at t=9×106MCS.

Fig.3 Snapshots of the evolution of nanoparticle/A1B3core-shell structures at different time10-6t/MCS:(a)0.1,(b)1,(c)2,(d)4,(e)7,(f)9;εnB=-0.9.To clearly see the core-shell structure,we shift 7 lattices along z direction in(f).

Fig.4 (a)Evolution of the system energy(E)during the formation of nanoparticle/A1B3core-shell structure,(b)variation of the densities(ρ)ofAbead,B bead,nanoparticle,and solvent with the distance(r)to the mass centerThe snapshots of a-f are presented in Fig.3.

Fig.5 Snapshots captured at different time10-5t/MCS:(a)0,(b)0.1,(c)1,(d)5;εnB=-1.6

Fig.4(a)presents the evolution of system energy during the formation of core-shell structure.At early time t＜106MCS,the system energy decreases quickly because nanoparticles rapidly form small aggregates followed by the fast growth of aggregates.During a long time interval from 1×106to 6×106MCS, the system energy varies little but the configuration changes obviously.During this period there is a competition between the following two tendencies:(1)The shrink of vesicle causes an increase in energy;while(2)the adsorption of A1B3chains on nanoparticle aggregates decreases the system energy.Their competition leads to the fluctuation of the system energy and the configuration evolution.However,the energy decreases gradually from 6×106MCS,indicating that the second tendency becomes dominating and the vesicle becomes smaller and smaller.At t=8×106MCS,A1B3vesicle disappears and the coreshell structure is formed instead,and the system reaches an equilibrium.

Fig.6 (a)Time evolution of the system energy E during the formation of nanoparticle-dispersed vesicle,(b)variation of the densities ofA, B beads,nanoparticle,and solvent with the distance(r)to the mass center of vesicleThe snapshots of a-d are presented in Fig.5.

The core-shell structure is characterized using the density distributions of different components with respect to the mass center,as shown in Fig.4(b).It shows that the core is occupied by nanoparticles within r＜8.The core size is comparable to the nanoparticle aggregate formed in the absence of block copolymer(Fig.1(b)).The peaks of A and B beads locate at about 10 and 9,respectively,indicating that A1B3chains form the shell with A bead on outer surface of the core-shell structure.The shell is fabricated by one layer of copolymers.

We have also investigated the evolution of nanoparticle-dispersed vesicle structure at εnB=-1.6(Fig.2(e)).Snapshots for the loading of nanoparticles in the vesicle shell are shown in Fig.5.Initially,nanoparticles are randomly put into the simulation box(Fig.5(a)).They quickly assemble into small aggregates due to their hydrophobic property.Meanwhile some nanoparticles diffuse into the vesicle shell owing to the attraction between nanoparticle and B bead,see Fig.5(b).Number of nanoparticle aggregates decreases with the time,more and more nanoparticles diffuse into the vesicle shell,as shown in Fig.5(b,c).At time t=1.5×105MCS,all nanoparticles are located in the vesicle shell.Fig.5(d)shows the snapshot captured at t=5×105MCS.This structure remains stable in the following 15×105MCS simulation.During the evolution,the system energy E drops monotonously(Fig.6(a)),indicating the evolution is determinatively controlled by energy.E reaches an equilibrium value at t～1.5×105MCS.Afterwards,the system energy roughly remains constant,indicating that the final structure is stable.

In order to obtain the detailed information on the final structure,the density distributions of different components with respect to the mass center of the vesicle are plotted in Fig.6(b). The vesicular core is occupied by solvent within r＜3.A beads locate at inner and outer surfaces of the vesicle showing two peaks at r～4.5 and～11.5 in the density profile,respectively.B beads form the vesicle shell.The shell thickness is about 7,bigger than that of the initial A1B3vesicle(Fig.1(a)).The peak position of nanoparticle distribution is symmetric with respective to the center of the shell,while the density profile of B beads shows two peaks separated by nanoparticles,suggesting that nanoparticles localize in the center of the vesicle shell.The result is consistent with the experimental report.16

4 Conclusions

In this work,the structures of nanoparticle/copolymer mixed system were studied using lattice dynamic Monte Carlo simulation.The amphiphilic property of A1B3chain is represented by an attraction between hydrophobic B-B beads,while nanoparticle is hydrophobic with attraction among nanoparticles.A1B3chains with εBB=-1 form a vesicle in solution at φp=7%,while nanoparticles with εBB=-1.5 assemble into a compact aggregate at φn=3%.In the simulation,nanoparticles are added into the vesicle solution.By tuning the attraction εnBbetween nanoparticle and B bead,we obtain two interesting structures:nanoparticle/A1B3core-shell structure and A1B3vesicle with nanoparticles dispersed in the shell.They are in agreement with experimental findings.

The evolutions of these two structures were investigated. The core-shell structure is developed through destroying the vesicle.While for the nanoparticle-dispersed vesicle,A1B3vesicle acts as a template and nanoparticles diffuse into the hydrophobic shell.The formation of the core-shell structure is a very time-consuming process,since it has to overcome an energy barrier to destroy the vesicle.Therefore,it is much slower than nanoparticle-dispersed vesicle.Our simulation provides a new way to load hydrophobic nanoparticles into A1B3micelle through breakingA1B3vesicle.

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Monte Carlo Simulation on the Structures of a Nanoparticle/ Copolymer Mixed System

LIU Guan-Feng HUANG Jian-Hua*
(Department of Chemistry,Zhejiang Sci-Tech University,Hangzhou 310018,P.R.China)

The structures of a nanoparticle/copolymer mixed system were studied using lattice dynamic Monte Carlo simulations.Each copolymer chain consisted of one A bead and three B beads,and the amphiphilic property of the A1B3chains was represented by an attraction between B-B beads. Nanoparticles were hydrophobic with an attraction amongst themselves.By properly choosing the attraction between the nanoparticle and the B beads,we observe two interesting structures:a nanoparticle/ A1B3chain core-shell structure and an A1B3vesicle with nanoparticles dispersed in the hydrophobic shell. The evolutions of these two structures were investigated.Our results show that the A1B3vesicle acts as a template for the formation of the nanoparticle-dispersed vesicle.

Dynamic Monte Carlo simulation;Copolymer;Nanoparticle;Vesicle

10.3866/PKU.WHXB201111211 www.whxb.pku.edu.cn

*Corresponding author.Email:jhhuang@zstu.edu.cn;Tel:+86-571-86843233.

The project was supported by the National Natural Science Foundation of China(21171145)and Natural Science Foundation of Zhejiang Province, China(Y4110422).

国家自然科学基金(21171145)和浙江省自然科学基金(Y4110422)资助项目

O641;O648