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Layer-by-layer assembly of colloidal particles deposited onto the polymer-grafted elastic s

a r X i v :c o n d -m a t /0611105v 1 [c o n d -m a t .s o f t ] 4 N o v 2006

Layer-by-layer assembly of colloidal particles deposited onto the polymer-grafted

elastic substrate

Kang Chen and Yu-qiang Ma ?

National Laboratory of Solid State Microstructures,Nanjing University,Nanjing 210093,China We demonstrate a novel route of spatially organizing the colloid arrangements on the polymer-grafted substrate by use of self-consistent ?eld and density functional theories.We ?nd that grafting of polymers onto a substrate can e?ectively control spatial dispersions of deposited colloids as a result of the balance between colloidal settling force and entropically elastic force of brushes,and colloids can form unexpected ordered structures on a grafting substrate.The depositing process of col-loidal particles onto the elastic “soft”substrate includes two steps:brush-mediated one-dimensional arrangement of colloidal crystals and controlled layer-by-layer growth driven entropically by non-adsorbing polymer solvent with increasing the particles.The result indicates a possibility for the production of highly ordered and defect-free structures by simply using the grafted substrate instead of periodically patterned templates,under appropriate selection of colloidal size,e?ective depositing potential,and brush coverage density.

Colloidal particles can self-assemble into a rich variety of highly ordered structures[1]on periodically patterned substrate [2],block copolymer sca?olds [3],vesicle sur-faces with the opposite charge [4],and at brush/air in-terfaces [5,6]and liquid/liquid [7]or water/air [8]in-terfaces.The major challenges in this ?eld are how to assemble monodispersed colloids into highly ordered structures with well-controlled sizes and shapes,and how to achieve layer-by-layer growth of ordered struc-tures.Sedimentation is a simplest approach for col-loidal crystallization,however,usually leads to uniform or simple close-packed arrays of colloidal particles on smooth substrates.E?ective control over interaction and arrangement of colloids[6,9]is possible by using densely polymer-grafted substrates[10],since the entrop-ically elastic energy of brushes is comparable to thermal energy,and self-assembly depends critically on thermal energy [4].A balance between depositing force of col-loids and entropic force of brushes probably leads to the formation of ordered colloidal crystal structures,rather than highly disordered aggregates.

The grafting of polymers to surfaces is a simple and useful approach to stabilize colloids against aggregation and adsorption.It was experimentally reported [11]that the thickness of brushes can be adjusted between several nm and 1μm .The grafted polymer always exerts a repul-sive entropic force on incoming particles,and past works focused on the interaction between polymer brushes and individual incoming particles and how to prevent the ad-sorption of colloids such as proteins onto surfaces under various grafting density,chain length,and interactions between chains and surface[10].To the best of our knowl-edge,there have been no systematic theoretical studies into the self-assembling structures of colloidal particles when deposited onto the grafting substrate.Here,we undertake the ?rst theoretical study of deposition of col-

2a 2

ds |d r α(s )

potential per solvent segment and Z n

α,nβ,n p

is the canon-

ical partition function for nαgrafted chains,nβsolvent polymers,and n p particles:

Z n

α,nβ,n p

n p!

nα

α=1

D rα(s)P[rα(s)]

nβ

β=1

D rβ(s)

P[rβ(s)]

n p

p=1

d R pδ[1? ?? ?s? ?p]

exp[?

ρ0k B T V =?φ(

ρ0V

e NμQ S?

Nφp Vφp

Nν

d r[ξ(1????s

??p)+w?+w s?s+w pρp?ρpΨ(

N |r′|

6?2q i?w i(r)q i,and q?i meets the same di?usion equa-

tion but with the right-hand side multiplied by?1.The last term in Eq.(2)is DFT term[16]accounting for the steric interaction between particles,and the excess free energyΨ(

?p(r)

[13].The interaction energyνis given by Nν/ρ0k B T=

d r[χbs N??s+χbp N??p+χsp N?s?p+g

e z?p],where theχ’s are the Flory-Huggins interaction parameters be-tween the di?erent chemical species.We?x N=100,χbs N=0,andχbp N=χsp N=12.0,since we assume that all the polymers have the same chemical nature, while the particles are insoluble to polymers[18].In ad-dition,a depositing force is applied normal to the

sub-

(f)

0.4500

0.9000

FIG.1:Particles density distributions in x-z cross-sections under di?erent particle volume fractions.(a)φp=0.03,(b)φp=0.04,(c)φp=0.05,(d)φp=0.06,(e)φp=0.092,(f)φp=0.1,and(g)φp=0.145.The color scale bar shows the local density values of particles in Figs.1(a)-1(g).

strate,and g e>0is the strength of settling?eld act-ing on particles.Here,we choose g e within the range 0.1～0.6which,on one hand,can ensure that the size of particles(2R=0.5R0)is comparable to the sedimen-tation length(∝N

d c

λc

FIG.2:(a)The minimum spacing d c between cylinders vs g e with error bars.(b)The critical correlation factor λc and penetration depth εc as a function of g e .

the smooth substrate.As more colloidal particles are deposited,the grafted polymer is compressed under de-positing potential,and it responds with a restoring force which further drives particles to self-assemble into cer-tain structures for counteracting deposition of particles,in the requirement of minimizing combinational contri-bution of colloidal depositing energy and brush entropy.Therefore,colloidal particles assemble into colloidal crys-tals under sedimentation,and morphology of regularly separated cylinder structures emerges.The number of cylinders will increase with the volume fraction φp (Figs.1(b)-1(d)).However,further increase of φp leads to the formation of second layer of cylinders(Figs.1(e)and 1(f))piled on the ?rst layer created by brushes,and even forms the third layer of cylinder structures(Fig.1(g)).

The ?rst-layered colloidal structure is formed,due to the competition between colloidal sedimentation and brush entropy e?ects.For a ?xed g e ,the deposited num-ber of cylinders of radius R increases with increasing φp when the variation of depositing potential per cylinder g e επR 2k B T/N ≥?F b .Here,?F b represents a brush stretching energy penalty of a local

single cylinder with the penetration depth εinto the brush [24],which is given by ?F b =1

3Nσ2ε,

and the equality is valid at the criti-cal φp where the minimum spacing between cylinders is reached,and larger φp will lead to second-layer aggre-gation of particles.From the SCFT/DFT calculations,we determine the minimum spacing d c (Fig.2(a))and the critical penetration depth εc of cylinders(Fig.2(b)),and ?nd that d c decreases,whereas εc increases with g e .This clearly shows that with increasing g e ,depositing po-tential of particles is balanced by further deformation of brushes.Figure 2(b)also gives the critical modi?ed fac-tor λc as a function of g e ,showing that at critical φp ,

0102030

4050600.0

0.20.40.60.8

1.0(a)

S s /k B

φp

FIG.3:(a)Lateral statistical concentration morphology of brush(dotted curve),free chains(solid curve),and parti-cles(dashed curve)for φp =0.1.(b)Entropies of brushes (S b )and solvent (S s )vs φp .

the e?ective shear modulus ?rst increases with increas-ing g e and then slightly decreases at large g e .For small g e ,the shear modulus increases with g e ,indicating that the correlation between cylinders is enhanced due to the decrease of d c .In contrast,for large g e ,the minimum spacing between cylinders keeps almost unchanged,but the embedding depth εc becomes larger than the cylinder radius so that the grafted chains can cross the narrow gap between cylinders and ?ll the upper space.Thus,the grafted chains are slightly released,which accounts for small decrease of λc at large g e (>0.5).

The formed ?rst-layer colloidal crystal may serve as a template for next-layer structural formation,and thus provides a possible route to the fabrication of multi-layer microstructures.We ?nd from Fig.1(f)that the second layer is well arranged,based on the already deposited layer.However,the morphology selection of second-layer colloidal dispersions will be a?ected by the entropic e?ect of solvent polymers.By calculating the z -direction aver-aged density pro?les of brushes,free chains,and particles for the case of Fig.1(f),Fig.3(a)clearly shows that more solvent chains will ?ll the space between the ?rst layer and the second layer of cylinders.Therefore,the second-layer cylinder structure is selected to alternately arrange with the ?rst layer for increasing the con?gurational en-tropy of con?ned solvent chains.In fact,the alternating arrangement of cylinders in Fig.1(g)further supports our viewpoint on controlled layer-by-layer growth driven entropically by polymer solvent.Figure 3(b)gives the entropies of brush and solvent as a function of φp .We ?nd that when φp takes the range 0.09～0.10corre-sponding to the forming process of second-layer struc-ture,the brush entropy retains almost unchanged,while the solvent entropy sharply declines,meaning that the second-layer colloidal assembly is out of brush e?ects,in-stead the solvent entropy dominates the ?nal equilibrium dispersion of the second-layer particles.

Finally,Fig.4shows the entropy of brushes and the root-mean-square ?uctuation ?of statistical brush height h [24]as a function of g e for φp =0.06and 0.1.We see that the brush entropy decreases with an increase of g e ,but the height ?uctuation due to brush deformation increases with g e .Correspondingly,the particle distribu-

g e

S b /k B

FIG.4:The brush entropy S b and the modulated ?uctuation ?of brush height vs g e .(a)φp =0.06.(b)φp =0.1.

tions in the inset of Fig.4signify the formation of cylin-der structures with varying g e .Figure 4(a)shows that for small g e ,the deposition of particles did not deform brushes which therefore do not react with the dispersion of particles.On the other hand,the relatively large range of the parameter g e can stabilize the one-layer cylinder structures due to strongly entropic restoring forces of brushes.In contrast,Fig.4(b)shows that a small range of g e may retain the two-layer cylinder structures.When g e is small,there are not enough particles deposited onto the top of brushes,leading to one-layer structure.As g e is relatively large,the two-layer structure is destroyed,in-stead the alternating structure of one-and bi-layer cylin-ders appears,because the entropy restoring force of sol-vent chains is weaker than that of brushes,and may not completely o?set the settling energy of particles if the

second layer is formed.It is actually interesting that the brushes have large entropic restoring forces which easily stabilize colloidal dispersions.For example,depending on the colloidal weight and volume fraction,brushes can adjust the number of cylinders formed,in contrast to col-loidal crystallization in non-adsorbing polymer solvent.In summary,we have demonstrated that under suitable density and depositing force of particles,colloidal par-ticles can be sorted into alternating arrays of cylinders by use of grafted substrates.The colloidal dispersions are dominated by the requirement of minimizing com-binational contribution of depositing potential of parti-cles and entropic restoring force of the deformed brushes.With an increase of colloidal additions,controlled layer-by-layer growth is driven by entropic e?ects of solvent chains.The advantage of the present approach is that control over arrangement of colloids did not rely on other patterned [2]and phase-separated copolymer [3]tem-plates but was achievable via a polymer-grafted substrate which is easily manufactured.The approach that under sedimentation,polymer entropic restoring force drives or-dered structure formation,will o?er a simple and power-ful alternative for producing 2D and even 3D structures,and may open up an unexplored route for engineering highly ordered structures from colloidal building blocks.This work was supported by the National Natural Sci-ence Foundation of China under Grant Nos.10334020,10021001,and 20490220.

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