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Nanoscale
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Cite this: Nanoscale, 2016, 8, 15961
Received 23rd June 2016,Accepted 30th July 2016
DOI: 10.1039/c6nr05043c
www.rsc.org/nanoscale
Chiral selection of single helix formed by diblockcopolymers confined in nanopores
Hanlin Deng,a Yicheng Qiang,a Tingting Zhang,b Weihua Li*a and Tao Yang*b
Chiral selection has attracted tremendous attention from the scientific communities, especially from bio-
logists, due to the mysterious origin of homochirality in life. The self-assembly of achiral block copolymers
confined in nanopores offers a simple but useful model of forming helical structures, where the helical
structures possess random chirality selection, i.e. equal probability of left-handedness and right-handed-
ness. Based on this model, we study the stimulus-response of chiral selection to external conditions by
introducing a designed chiral pattern onto the inner surface of a nanopore, aiming to obtain a defect-free
helix with controllable homochirality. A cell dynamics simulation based on the time-dependent Ginzburg–
Landau theory is carried out to demonstrate the tuning effect of the patterned surface on the chiral selec-
tion. Our results illustrate that the chirality of the helix can be successfully controlled to be consistent with
that of the tailored surface patterns. This work provides a successful example for the stimulus response of
the chiral selection of self-assembled morphologies from achiral macromolecules to external conditions,
and hence sheds light on the understanding of the mechanism of the stimulus response.
Chirality, or called dissymmetry, is an unique feature of anobject that cannot be superposed onto its mirror image.1,2
Chiral objects in the form of small molecules or large assem-blies are ubiquitous in nature as well as in manmadematerials, and thus have attracted tremendous attention fromthe scientific communities of biology, chemistry, physics, andmaterials.3–5 In biology, amino acids, as one of the mostimportant constituents, of life are chiral and nearly all of thosein proteins are left-handed, while all sugars in DNA and RNAare right-handed.6,7 Moreover, there are many helical struc-tures in the form of chiral assemblies in biological systems,such as α-helices in proteins, DNA double-strand helices, ten-drils, and the coiled shells of gastropod species.8–10 In particu-lar, the origin of homochirality in life remains mysteriousthough many efforts have been devoted to this topic.6,11,12 Incontrast, the chemical synthesis of chiral compounds fromachiral reagents always yields a racemic mixture of equally rep-resented enantiomers.13 From the perspectives of both funda-mental research and the technological applications, thispresents an interesting challenge to control the imparted chir-ality of molecules or helical structures. In experiments, diverse
external chemical and physical chiral conditions have beendevised to induce chiral selection.5,14–19 However, the induc-tion process of chiral control has been much less explored bytheory.20–22
Block copolymer self-assembly has attracted abiding inter-est because it provides a powerful platform for the formationof ordered nanostructures which have potential applications ina wide range of fields.23–25 In particular, the self-assembly be-havior can be readily enriched by the diverse molecular archi-tectures of block copolymers as well as its facile stimulusresponse to external conditions. It has been established thatthe simplest bicomponent (or AB-type) block copolymer, ABdiblock copolymer, can self-assemble into a set of orderedphases including lamellar, cylindrical, spherical, or gyroid (orFddd) networks.26,27 Although the structure library from theself-assembly of bicomponent block copolymers cannot beexpanded dramatically, a few new phases have been stabilizedin AB-type multiblock copolymers with various topologicalarchitectures.27–29 For example, the perforated lamellar mor-phology that is usually metastable in simple AB diblock or ABAtriblock copolymers becomes stable in comb-like or branchedAB-type block copolymers.27 More interestingly, a few non-classical phases, including square cylinders, graphene-likecylinders, and honeycomb-like networks, have been predictedby theory in AB-type block copolymers with purposelydesigned miktoarm architectures following useful guidingprinciples.29 Moreover, a large number of new structures inthe form of binary mesocrystals have been obtained from theself-assembly of designed B1AB2CB3 multiblock terpolymers.30
aState Key Laboratory of Molecular Engineering of Polymers, Department of
Macromolecular Science, Fudan University, Shanghai 200433, China.
E-mail: [email protected]; Fax: +86 (0)21 65640293; Tel: +86 (0)21 65643579,
+86 (0)951 2061707bNingxia Key Laboratory of Information Sensing & Intelligent Desert, School of
Physics & Electrical Information Engineering, Ningxia University, Yinchuan 750021,
China. E-mail: [email protected]
This journal is © The Royal Society of Chemistry 2016 Nanoscale, 2016, 8, 15961–15969 | 15961
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As a typical soft matter system, the self-assembly of blockcopolymers has a strong responsive capability to external con-ditions, e.g. the confinement imposed by geometrical bound-aries.31 The geometrical confinement not only breaks thetranslational symmetry of the periodic phases in the bulk butalso alters the domain geometry, leading to a lot of intriguingnovel morphologies differing from the bulk phases.32–40 Oneof the most extensively studied geometrical confinements iscylindrical confinement. Many attractive morphologies havebeen observed in the self-assembly of block copolymers con-fined in nanopores by experiments,32–34 computer simu-lations35,36 and theoretical calculations.37–40 One type ofsurprising morphology is the helical structure from the self-assembly of achiral block copolymers under cylindrical con-finement. The fundamental formation mechanism has beenestablished; the mismatched lengths between the nanoporeand the cylinder, due to a preferred diameter of the latter for agiven block copolymer, drive the longer cylinder to curve intohelices.35,37 Similarly, helical structures can be self-assembledby linear and star triblock copolymers softly confined bysolvent environments.41,42 In addition to the self-assembly ofblock copolymers, the cylindrical confinement can also inducethe formation of helical structures in small molecule systems.For instance, nanotubes (e.g. carbon nanotube) have beenapplied to template the assembly behavior of surfactants43 orwater molecules.21,44 In particular, a variety of helical struc-tures have been predicted by molecular dynamics simulationsin the high-density nano-ice confined in carbon nanotubes.21
Due to the intrinsic achiral property of the block copolymersystems, the formation of helical structures possesses arandom selection of chirality, i.e. equal probability of left-handedness or right-handedness.35,37 Similar to the racemicmixture in chemically synthesized chiral compounds wherethe enantiomers have identical free energy, the left-handedand right-handed helices are degenerate in free energy.Besides the cylindrically confined system, this phenomenonhas also been observed with bulk achiral block copolymersystems, such as in the self-assembly of linear ABC triblockcopolymers in helical supercylindrical structures.45 For a cylin-drically confined system that is quasi-one-dimensional, helicalstructures with a random chirality are spontaneously formedat any position along the axial direction of the nanopore, andthus usually exhibit nonuniform chirality in a macroscopicallylong nanopore. As a result, defects are unavoidably caused atthe interfaces between the morphological pieces with differentchirality, thus destroying the one-dimensional long-rangeorder. A variety of defects have been explored in confinedsystems with a large length by cell dynamics simulation (CDS)based on time-dependent Ginzburg–Landau theory (TDGL).46
Note that the loss of long-range order in the one-dimensionalsystem is robust for one-dimensional condensed mattersystems, which originates from the dominant effect of theentropy over the interfacial energy or the defective energy.
It is apparent that helical structures with controllable anduniform chirality are more likely to satisfy practical appli-cations. How to control the chirality of the helical structures by
the self-assembly of cylindrically confined block copolymerspresents an interesting question. A number of strategies havebeen proposed in experiments to control the chiral selection ofhelical structures formed by self-assembled systems.14–19 Forexample, Yang and coworkers utilized chiral templates andchiral terminals, respectively, to induce the formation ofhelical structures with desired chirality in various self-assem-bly systems without chiral selection.14,15,17 More instructively,Yao et al. added small chiral molecules to regulate the self-assembly of helical superstructures with controllable chiralityfrom achiral diblock copolymers, which is a typical stimulusresponse process.16 However, this interesting topic is rarelyexplored by theory due to the vast challenge in methodology.On the one hand, an appropriate theoretical model is neededto simulate the self-assembly process of helical structures thattypically involves nano- or mesoscale system sizes. On theother hand, the model must be capable of capturing the chiralnature of the external environment at the molecular level.
In this work, we take the single-helix morphology formedby the cylindrically confined diblock copolymers as anexample model to study the stimuli-response of chirality ofhelical structures to external conditions. Instead of dealingwith the chiral conditions at the molecular level, we simplydesign a chiral wall surface of the nanopore to mimic the exter-nal chiral conditions, thus investigating the induction effect ofthe chiral confining conditions on the chirality of the singlehelix inside the nanopore. The pore surface is patterned with aset of correlated patches with a distinct surface potential fromthe non-patched area, which is characterized by the size,shape and distribution of the patches as well as the contrast ofthe surface potential between the patched and non-patchedarea. Our main goal is to optimize the surface pattern for thecontrolled selection of chirality of the helical structure by sys-tematically simulating the formation kinetics of the singlehelix under the confinement with variable chiral surfacesusing the TDGL theory, and accordingly to unveil the stimulusresponse of the chirality of the helical structures to externalchiral environments. The CDS of TDGL theory has beenproven to be a high-efficiency dynamic simulation method forthe simulation of the collective kinetics of block copolymerself-assembly.47–50
Model and theory
We consider an incompressible melt of asymmetric AB diblockcopolymers with a fixed volume fraction of the A block f =NA/N, where NA and N are the polymerization degrees of theA block and the entire polymer chain, respectively, confined ina cylindrical nanopore of diameter D. The order parameter, asa spatial function that characterizes the phase separation ofthe diblock copolymer melt, is chosen as ϕ(r) = ϕA(r) − ϕB(r),where ϕA(r) and ϕB(r) are the compositional distributions ofthe A and B components. Under the mean-field approxi-mation, the free energy of this system can be expressed as afunctional of ϕ(r), which is composed of three contributions,
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short-range, long-range, and the surface energy of the porewall:50
F½ϕ� ¼ FS½ϕ� þ FL½ϕ� þðrj j,D=2
drHsurfðrÞϕðrÞ; ð1Þ
where the integration is restricted inside the nanopore. Theshort-range term FS is the usual Ginzburg–Landau type freeenergy and can be written as
FS½ϕ� ¼ðrj j,D=2
drC2
∇ϕðrÞj j2 þWðϕÞ� �
; ð2Þ
where the first term with a positive constant C is applied tosuppress the short wavelength fluctuations, and the secondterm W(ϕ) is the local interaction contribution and it can beconveniently specified by its derivative form,
dWðϕÞdϕ
¼ �A0 tanhðϕÞ þ ϕ: ð3Þ
The above constant A0 > 1 gives rise to the phase separationbetween the two immiscible components. The long-range con-tribution is originally derived by Ohta and Kawasaki from thesecond-order vertex function obtained by the random phaseapproximation to alter the phase separation from macroscopicin A/B blends to microscopic in AB diblock copolymers,51
which is expressed as
FLðϕÞ ¼ α
2
ðdr
ðdr′Gðr� r′ÞδϕðrÞδϕðr′Þ; ð4Þ
where δϕ(r) = ϕ(r) − ϕ̄ and ϕ̄ = 2f − 1 is the spatial average ofϕ(r). In the above expression, the positive constant, α, isinherent to the characteristics of the block copolymer, andthereby dictates the segregation degree as well as the domainspacing. The long-range feature stems from the Green functionG(r − r′) because it shares the Coulomb potential form. Inpractice, the Laplacian equation of the Green function is moreconvenient, which is
�∇2Gðr� r′Þ ¼ δðr� r′Þ: ð5Þ
In the last term of eqn (1), the spatial function of Hsurf(r) isimplemented to mimic the preferential interaction potentialsof the pore wall on the two species. In order to include thechiral property of the pore surface, Hsurf(r) has to be modifiedfrom a simple function of the radial distance r to be a morecomplex function of not only r but also the polar angle θ andthe z coordinate along the pore axis. First, we define a generalsurface potential for the entire pore surface, Hpore(r), as
HporeðrÞ ¼ 12V0ftanh½ðσ � dðrÞÞ=ε� þ 1g ð6Þ
where d(r) is the distance to the boundary along the radialdirection, and σ and ε specify the interaction distance andshape of the external potential along the radial direction,respectively. The constant V0 gives the potential strength of thenon-patched area.
The pore surface is then patterned via introducing Np corre-lated patches onto the surface, whose distribution ischaracterized by a set of positional parameters (hi, θi) (i = 1, 2,…, Np), where hi and θi indicate the axial position and thepolar angle of patch i, respectively (Fig. 1(b)). In practice, thepattern of each patch is realized by using an elliptic cylinderwith the major axis 2Rp and the minor axis Rp to “cut” thenanopore, thus dividing the surface potential into the patchedand non-patched area. In other words, the projection of eachpatch along the radial direction is an ellipse with the majorand minor axes as 2Rp and Rp, respectively. The surface poten-tial in the patched area delimited by the elliptic cylinder is setto be distinguished from that of the non-patched area withHsurf = Hpore by a modification factor, k, i.e. Hsurf = kHpore. Notethat the surface potential is not ideally restricted on the poresurface, but it is indeed extended into the pore due to thecertain interaction range.
With the above explicit free-energy functional, the phase-separation dynamics of the block copolymer system can bedescribed by the Cahn–Hilliard dynamic equation (Model B),
@ϕ
@t¼ M∇2 δF½ϕ�
δϕþ ζðr; tÞ; ð7Þ
where M is relevant to the chain mobility, set to M = 1, and ζ(r,t )is a random noise term with zero average and a second momentof ⟨ζ(r,t )ζ(r′,t′)⟩ = −ζ0M∇2δ(r − r′)δ(t − t′); here ζ0 is the noisestrength.
We choose the group of parameters to be f = 0.38, A0 = 1.28,and α = 0.02, to obtain a typical hexagonal cylinder phase in
Fig. 1 (a) Schematic plot of the self-assembly system of cylinder-forming AB diblock copolymers under the confinement of a nanoporewith length l and diameter D. The cylinder-to-cylinder distance of thebulk phase of the diblock copolymer is indicated by L0. The pore surfaceis patterned with a set of patches, which are chirally and periodically dis-tributed along the pore and thus have correlated positions (hi, θi). Theshape of each patch is specified by its projection along the radial direc-tion that is an ellipse with the lengths of the major and minor axes as2Rp and Rp, respectively, and with the minor axis aligning along the poreaxis. (b) Fluctuating proportion of the right-handed helix of all helicalmorphologies formed in the nanopores with a non-patterned surfaceand fixed diameter D = 1.74 L0 for a varying number of running samples.The helical pitch is indicated by d0. NRH and NLH indicates the number ofleft-handed and right-handed helices, respectively.
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the bulk, and specify these parameters of the surface potentialas V0 = 0.015 (for majority-attraction), σ = 0.15 L0, and ε = 0.5L0 (L0 is the cylinder-to-cylinder distance in the bulk cylinderphase). The standard CDS method, where the forward Euleralgorithm is applied for the time integration and the explicitfinite differential scheme is applied for the space, has beenproven to be efficient to solve the TDGL equations.50 In oursimulations, the lattice spacing in the CDS simulations is fixedas Δx = Δy = Δz ≡ Δ = 0.50, and accordingly the time step Δt ischosen as 0.1 to ensure the solution stability of the dynamicequations. L0 is determined to be about 16.5 Δ by the simu-lation of the bulk system. Periodic boundary conditions areimposed on the z direction. For reading convenience, the mainparameters are summarized in the Appendix.
Results and discussion
The self-assembly of cylinder-forming AB diblock copolymersunder cylindrical confinement has been examined previouslyusing CDS simulations.46 The kinetically stable region ofvarious morphologies, including single cylinder, stackeddisks, single helix, and double helices, have been determinedwith respect to the pore diameter. Accordingly, we fix the porediameter as D = 1.74 L0, which is located almost at the centerof the phase region of the single helix, to ensure the formationof single helix during the kinetics. In principle, the structureformation in this quasi one-dimensional system is sensitive tothe pore length with a periodic boundary condition at twoends. Only when the pore is short ,such as a few L0, a singlehelix with a uniform chirality is likely to be formed, but thechiral selection of each independent run of the dynamic simu-lations is random between the left-handed and right-handedtypes. Otherwise, the morphologies are formed in a long pore,typically with a length of tens of L0, consisting of helical pieceswith mixed left-handed and right-handed chirality, thus exhi-biting defects at the interface of different chiralities. Thisimplies two equivalent examination strategies for the chiralselection of single helix in the currently considered system.One strategy is to statistically count the probability of a singlehelix with a given chirality from a large number of simulationruns on the cylindrically confined system in a short nanoporewhere the single helix of a uniform chirality is mainly formed.The other is to determine the proportion of the single-helicalmorphology with a given chirality formed in a macroscopicallylong pore. In this work, we first adopt the former strategy toinvestigate the tuning effect of the designed patterns on thechiral selection, hence fixing the pore length as l = 9.68 L0,with a nanopore in which a helix of five pitches is formed as l≈ 5 d0 and d0 ≈ 1.94 L0.
In Fig. 1(b), the proportion of the right-handed single helixformed in the nanopores with a non-patterned surface isshown for various numbers of running samples. Obviously, itfluctuates and the fluctuation amplitude attenuates as thesamples are added, finally approaching 50%, i.e. an equalchance of selection for the right-handed and left-handed
helices. Then we start with a simple case of patterned surface,in which the patches with a fixed size of Rp = 0.25 L0 are uni-formly distributed in the entire pore and their polar angles areassociated with their positions forming a right-handed chiraldistribution. The patch distribution is specified by hi = (i −1/2)l/Np and θi = 2πi/Np (i = 1, 2,…, Np) with Np = 4 (Fig. 1(a)).Here we refer to this distribution as the uniform distribution.The proportion of the right-handed helix is estimated withsample numbers 100 and 500, respectively, and is shown inFig. 2. The large sample number of 500 leads to results withreasonably low statistical errors. Note that there is still someprobability for the system to form imperfect helical mor-phologies in such short nanopores, which is also counted inour simulations and is presented in Fig. 2.
The modification factor k = 1 corresponds to a nanoporewith non-patterned surface that does not exhibit any chirality.In this case, the pore surface is uniformly attractive to themajority component. As a consequence, the formed helicalmorphology selects the left-handed and right-handed chiralitywith an equal probability of 50%. Moreover, the random selec-tion of chirality leads to a possibility as high as around 40% offorming defective morphologies composed of helical pieceswith different chirality (Fig. 2). When k is varied to be differentfrom 1, a right-handed chiral property is introduced to thepore surface due to the distinguished surface interaction ofthese patterned patches from the non-patterned area, thusleading to the selection of right-handed chirality. Note that theright-handed chiral feature is dictated by the contrast of thesurface potential between the patterned and non-patternedarea, which holds equally for the two cases of k > 1 and k < 1.As k deviates from 1 more and more, the proportion of theright-handed helix increases, which is accompanied by the
Fig. 2 Proportion of right-handed helical morphologies as well asdefective morphologies self-assembled by the diblock copolymersconfined in the nanopores with patterned surface and a given length l =9.68 L0, as a function of the modification factor k. The pattern is com-posed of four patches with size Rp = 0.25 L0 whose positions along thepore axis and associated polar angles are hi = (i − 1/2)l/4 and θ = iπ/2 (i =1, 2, 3, 4). Two groups of data for the sample numbers of 100 and 500are presented. NDM indicates the number of defective morphologies.
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decreasing proportion of defective morphologies. For k ≥ 1.4,the right-handed and defect-free helix is formed exclusively inthe nanopore. Obviously, the pattern with k < 1 is also able totune the chiral selection. However, small k, e.g. k < 0.8, willweaken the surface attraction to the majority component,accordingly raising the possibility of creating defects as a newmechanism differing from that arising from the uncontrolledselection of chirality. Therefore, we do not extend our simu-lation to the further end of k < 0.8.
To testify the tuning effect of the designed chiral pattern onthe chiral selection in an alternative way, we simulated theself-assembly of the system with a pore length triple that of theprevious one, i.e. l = 3 × 9.68 L0 ≈ 15 d0, for three typical valuesof the modification factor, k = 1.0, 1.2, and 1.4 (Fig. 3). Withthe non-patterned surface of k = 1.0, although a piece of right-handed helix (in green color) is formed in the pore, two piecesof other distinct morphologies are also formed, including theleft-handed helix and tilted toroids, interfering with theuniform formation of right-handed helix throughout the entirepore. In contrast, for the chiral pattern of k = 1.2, the pro-portion of the desired right-handed helix in the pore isenhanced significantly. Finally, with k = 1.4, the right-handedhelix is self-assembled throughout the nanopore, corres-ponding to a defect-free helical morphology. This observationis consistent with that from the results on the formation prob-ability of the right-handed helix in the short nanopores of l =9.68 L0. In brief, the designed pattern of patches on the poresurface with tailored chiral properties has been demonstratedto be able to guide chiral selection for the self-assembly ofhelical morphologies from achiral block copolymers confinedin the nanopores, and thus lead to the formation of defect-freehelical morphologies in the quasi one-dimensional system.
Besides the critical variable of the modification factor k, thechiral property of the surface pattern can also be regulated bya few other parameters, such as the patch size Rp, number Np,and distribution (hi, θi). In Fig. 4, three different sizes ofpatches are considered: Rp = 0.20 L0, 0.25 L0, and 0.30 L0 forthe same number and distribution of patches as in Fig. 2.
Obviously, the surface pattern with larger patches leads to theformation of a higher proportion of the right-handed helix fora given value of k smaller than the critical value at which 100%of the right-handed helix is formed, indicating a larger tuningeffect on the chiral selection. However, we can speculate thatthe patch size might have a limit for increasing the tuningeffect, which will be discussed later.
In above examples, we fix the number of patches Np =4. Now we attempt to change Np to examine its influence onthe tuning effect of chirality. In Fig. 5, the results on the pro-portion of the right-handed helix for three values, Np = 3, 4,and 5, are presented, where the uniform distribution of thepatches is considered. In contrast to the variable Rp, Np has amore remarkable impact on the tuning effect of chirality. Thetuning effect of Np = 5 becomes considerably worse than thatof Np = 4, even though the chiral property is strengthened.This observation implies that the tuning effect of chirality is
Fig. 3 Three morphological samples formed in the cylindricallyconfined block copolymer system with a large pore length of l = 29.04L0 for various values of the modification factor, k = 1.0, 1.2, and 1.4(from top to bottom). For reasons of clarity, the desired right-handedhelical pieces are shown in green color, while the other morphologicalpieces are plotted in a darker color. The nanopore is totally decoratedwith Np = 12 patches, which are distributed into mp = 3 pattern periodseach of which consists of 4 patches.
Fig. 4 Proportion of right-handed helical morphologies as well asdefective morphologies as a function of the modification factor, k, forthree patch sizes Rp = 0.20 L0, 0.25 L0, and 0.30 L0 with fixed Np = 4.
Fig. 5 Proportion of the right-handed helix as a function of k for threeuniform surface patterns with various patch numbers Np = 3, 4, and 5,but fixed patch size Rp = 0.25 L0.
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not simply dictated by the magnitude of the chiral property ofthe patterned surface. The chiral properties of the externalconditions have to be commensurate with those of the helicalmorphologies. More interestingly, for Np = 3, the low pro-portion of the right-handed helix reveals that the formation ofthe left-handed helix is more likely to be induced by a surfacepattern with the opposite chiral property. However, from thepoint of view of forming unique chirality, the tuning effect isalso worse than that of Np = 4 because there is still a smallportion of the right-handed helix when k = 1.5.
To probe the formation mechanism of the helical mor-phology with opposite chirality to the external guiding con-ditions, we focus on the case of Np = 3 and attempt to modifythe distribution of the patches aiming to regulate the tuningeffect of chirality for the formation of a single helix with a con-sistent chirality to the surface pattern. For the uniform distri-bution of Np = 3, we plotted the density isosurface of the right-handed helix in the patched nanopore in Fig. 6. It is shownthat two patches are located on the pore surface where thehelical domain passes by (denoted as 1 and 3), whereas theother is located on the surface between two neighboringpitches of the helix (denoted as 2). Accordingly, the strongersurface attraction to the majority imposed by patches 1 and 3is unfavored by the right-handed helix, thus leading to its lowformation probability, lower than 10% at k = 1.5, when thepatch distribution is altered to be nonuniform by movingpatches 1 and 3 in the uniform distribution upward and down-ward by a half pitch, respectively, such that the two patcheslike patch 2 are located on the surface between two pitches.The patch distribution can be specified by hi = (4i/15 − 1/30)land θi = 2πi/3 (i = 1, 2, 3). With this modification, the for-mation of the right-handed helix is reversed to be overwhel-mingly dominant with a proportion close to 100% at k = 1.5(circle symbols shown in red).
For the uniform distribution of the patches, the number Np
of the patches for an ideal tuning effect can be simply derivedfrom the pore length l in the unit of the helical pitch d0 by
abiding by the energetic principle that all patches are locatedon the surface between two pitches. Assuming that the chiraldistribution of Np patches possesses mp periods, the helical mor-phology can have j + mp/Np pitches in the intermediate spacebetween two neighboring patches, where j = 0, 1, 2,… is aninteger. Thus we can get l/d0 = ( j + mp/Np)Np = jNp + mp for anideal surface pattern with a uniform distribution of patches. Forexample, in Fig. 3, the pore length l/d0 = 15, the patch numberNp = 12 and the number of pattern periods mp = 3 give rise to j =1. In contrast, for l/d0 = 5, Np = 3, and mp = 1 in Fig. 5, no integeris available for j, thus leading to a poor tuning effect of consist-ent chirality with the surface pattern. Instead, an oppositelychiral helix is induced by this imperfect surface pattern.
Conclusions
In summary, we propose a model system to study the stimulusresponse of chiral selection to the external conditions in theself-assembly of helical structures from achiral macromolecu-lar systems. The model system is realized in diblock copoly-mers under the confinement of nanopores whose surface ispatterned by a set of patches with distinct surface interactionfrom the non-patched area. Furthermore, the positions ofthese patches are arranged to obtain a chiral distribution onthe pore surface. The tuning effect of the chirality of the pat-terned nanopore is demonstrated by CDS dynamic simulationson the self-assembly of left-handed or right-handed helicalmorphologies. Our results suggest that the self-assembly of thesingle helix from the cylindrically confined diblock copolymersystem exhibits a sensitive stimulus response of chiral selec-tion to the tailored surface pattern of the nanopore. The pro-portion of single helix with a consistent chirality to the chiralpattern, starting from 50% in the case of non-patterned nano-pores, increases until it reaches 100% as the magnitude of thechiral property of the guiding surface pattern is enlarged bythe characteristic parameters of the pattern, such as the modi-fication factor k that characterizes the contrast of the surfaceinteractions between the patterned and non-patterned areas,and the patch size Rp. Moreover, the number of patches as wellas their distribution has a subtle influence on the tuningeffect of the surface pattern on chiral selection. Surprisingly,the pattern with Np = 5 leads to a considerably worse tuningeffect than that of Np = 4. This observation implies that a com-mensurability is required between the guiding pattern and theself-assembled helical morphology, for instance, in the period-icity. More interestingly, the chirality of the single helixformed in the patterned nanopore with Np = 3 can be con-trolled to be consistent or opposite to that of the chiral patternby tailoring the distribution of the patterned patches. Byprobing the induction mechanism of the chiral pattern on thechiral selection of the helix, an ideal distribution of thepatches is derived.
The significance of our work is two-fold. On the one hand,it provides a simple but useful model system for studying thestimulus response of the chiral selection to external con-
Fig. 6 Proportion of the right-handed helix as a function of k for theuniform and nonuniform surface patterns with Np = 3 and Rp = 0.25 L0.The nonuniform distribution of the patches is specified by hi = (4i/15 −1/30)l and θi = 2πi/3 (i = 1, 2, 3). The right-handed helical morphologiesin the two cases are plotted. For the uniform surface pattern, the threepatches from bottom to top are numbered as 1, 2, and 3, respectively.
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ditions. Although, our system may exhibit technical difficultiesin the preparation of the surface pattern of nanopores, thistechnical problem will hopefully be solved as nanotechnologyadvances. For example, there is experimental research attempt-ing to fabricate chiral mesopores via the self-assembly ofchiral amphiphilic molecules.52–54 These mesopores, withtunable diameter and chiral period, may provide a useful tem-plate to guide block copolymers to self-assemble into singlehelical morphologies with desired chirality. Moreover, thechiral nanopore may be effectively made from chiral materials.For example, zigzag carbon nanotubes are chiral. Though thediameter of carbon nanotubes is too small for block copoly-mers, they should be able to induce small molecules (e.g. water)to form helical structures with a given handedness.21 Therefore,our results are instructive for experiments to obtain defect-freehomochiral helices in a macroscopically long nanopore thatmay have promising applications in nanotechnology. On theother hand, our results reveal that chiral interaction on the poresurface is transferred into the self-assembly of the entire systemand thus successfully induces the chiral selection of the helix.This conclusion for the stimulus response mechanism of chiralselection may hold robustly for the self-assembly of otherhelical structures from achiral constituents, and thus shed lighton the understanding of the origin of homochirality in life.Importantly, the precise conditions of the chiral pattern associ-ated with helical geometry is predicted in a quantitative mannerfor the induction of a homochiral helix.
Appendix
List of the main parameters
Parameter Physical meaning Values
f Volume fraction of the minorityblock of AB diblock copolymer
0.38
A0 Model parameter controlling thedegree of phase separation
1.28
α Model parameter dictating thesegregation degree as well as thedomain spacing
0.02
Δ Lattice spacing 0.5Δt Time step 0.1L0 Cylinder-to-cylinder distance in
the bulk16.5 Δ
d0 Helical pitch of the single helix 1.94 L0l Length of the nanopore 9.68 L0 (5 d0),
29.04 L0 (15 d0)D Diameter of the nanopore 1.74 L0V0 Strength of the surface potential
of the non-patched area0.015
σ Interaction distance of the surfacepotential
0.15 L0
ε Steepness of the surface potential 0.5 L0k Modification factor of the surface
potential of the patches relative tothe non-patched area
0.8–1.5
Np Number of the patches 3, 4, 5, 12Rp Size of the patches 0.20 L0, 0.25 L0,
0.30 L0mp Number of periods of the
surface pattern1, 3
Acknowledgements
W. H. L. acknowledges the funding support by the NationalNatural Science Foundation of China (NSFC) (Grants No.21322407, 21574026). T. Y. thanks the funding support byNSFC (Grant No. 11504190) and the Natural ScienceFoundation of Ningxia (Grant No. NZ1640).
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