Form Factor of Asymmetric Elongated Micelles: Playing with Russian

Article pubs.acs.org/JPCB

Form Factor of Asymmetric Elongated Micelles: Playing with Russian Dolls Has Never Been so Informative Gerald Guerin,† Graeme Cambridge,† Mohsen Soleimani,†,‡ Sepehr Mastour Tehrani,†,‡ Ian Manners,*,§ and Mitchell A. Winnik*,†,‡ †

Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, ON M5S 3H6, Canada Department of Chemical Engineering and Applied Chemistry, University of Toronto, 200 College Street, Toronto, ON M5S 3E5, Canada § School of Chemistry, University of Bristol, Bristol BS8 1TS, U.K. ‡

S Supporting Information *

ABSTRACT: Scattering techniques (i.e., static light scattering, small angle neutron scattering,11 or small angle X-ray scattering) are excellent tools to study nanoscopic objects in solution. However, to interpret the experimental data, one needs to use the appropriate form factor. While recent progress has been made in the writing of form factors for complex structures, there is still a need to develop a method to evaluate the form factor of inhomogeneous elongated scatterers. Here, we propose an approach based on the principle of “Russian dolls”. Multiblock rods are represented as multi generations of rods (mother, daughter, granddaughter, etc.), where each rod is nested within the rod of the previous generation, like Russian dolls. A shift parameter is used to introduce asymmetry in the rod along its long axis. This approach not only allowed us to write the form factor of multiblock rods, but it also gave us the possibility to account for the polydispersity in length of each block and of the shift parameter. Finally, we applied these equations to the case of a series of solutions of triblock comicelles slightly polydisperse in length.

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INTRODUCTION Miniaturization of complex devices has inspired the scientific community to develop new approaches to construct nanostructures with specific shape and size control.1,2 By synthesizing new molecules or introducing new self-assembly strategies, novel structures with well-defined architecture have become accessible.3,4 For example, Gröschel et al.5 were able to control the formation of complex multicompartment micelles using the directed hierarchical self-assembly of triblock copolymers. Similarly, Pochan et al.6 showed that multicompartment micelles can be formed using hydrogen bonding to confine a block copolymer in a well-defined architecture. The characterization of these new structures can be challenging, and they are studied mainly by electron microscopy (EM). While EM offers a relatively direct way to visualize the objects formed, one often needs to stain selectively the different blocks to distinguish them. Moreover, by EM, the structures are normally studied in the dry state, which may differ from the solvated state. To characterize these structures in solution, one has to rely on scattering techniques. More information, such as the weightaverage molecular weight, the aggregation number, and the size of the solvated nanostructure can be accessed when one uses the appropriate form factor to fit the scattered intensity. Svaneborg and Petersen have recently published a new formalism, which allows one to obtain form factors of complex structures.7,8 Their formalism is particularly useful when one studies micelles consisting of corona chains protruding from a uniform shape. Pickering © XXXX American Chemical Society

emulsions and spherical Janus particles can also be studied using the form factor developed by Eliçabe.9 Rodlike particles have been intensively studied by scattering techniques because of their abundance and their important role in biology.10 The form factor of elongated structures has been developed for different types of cross sections: for an infinitely thin rigid rod, for a thick rod with a circular or elliptical cross section, and for tubes and concentric cross sections.11 More complex elongated structures have also been studied. For example, Holtzer and Rice12 developed the form factor for thin rigid rods with inhomogeneous mass distribution along the rod main axis to evaluate the structure of myosin. Form factors for cylindrical Janus micelles and patchy micelles have also been developed.13−15 However, these form factors cannot be used to characterize newly formed asymmetric elongated structures, such as those presented in Scheme 1. For example, Schmelz et al.16 obtained mainly asymmetric diblock comicelles by the self-assembly of PS−PE−PS and PS−PE−PMMA triblock copolymers (where PS stands for polystyrene, PE for polyethylene, and PMMA for poly(methyl methacrylate)), when PS−PE−PS unimers were added to a solution of PS−PE− PMMA seeds, as shown in Scheme 1a. Using DNA tiles, Mohammed and Schulman were able to build “match-type” Received: March 20, 2014 Revised: August 8, 2014

A

dx.doi.org/10.1021/jp502806v | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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Scheme 1. Examples of Reported Asymmetric Elongated Structuresa

a

(a) Polystyrene-block-polyethylene-block-poly(methyl methacrylate) (PS−PE−PMMA) added to a solution of PS−PE−PS seeds preferentially grows from one end of the seeds to form patchy micelles (Schmelz and Schmalz16). (b) A heterojunction between two tubes can be formed using DNA seeds17 or using graphitelike semiconductor nanotubes.18 (c) Asymmetric triblock comicelles formed by the crystallization driven self-assembly of polyferrocenyldimethylsilane-containing block copolymers. Rupar et al., for example, prepared PFS-b-PI/PFS-b-PDMS/PFS-P2VP triblock comicelles (PI: polyisoprene; PDMS: polydimethylsiloxane; and P2VP: poly(2-vinyl pyridine).26

study of a solution of MPFS−PI-b-MPFS−PDMS-b-MPFS−PI triblock comicelles by static light scattering (SLS).

structures by growing DNA nanotubes unidirectionaly from a DNA origami seed (Scheme 1b).17 Other types of segmented nanotubes, made from the heterojunctions between graphitelike semiconductors, have also been reported by the group of Aida.18 Our group has studied the formation of elongated micelles from the self-assembly of polyferrocenyldimethylsilane (PFS) block copolymers (BCPs).19 PFS block copolymers with a corona-forming block longer than the PFS block has a strong propensity to form elongated micelles in solvents selective for the non-PFS block. This crystallization-driven self-assembly process allowed us to control the length and architecture of these PFS-based micelles. For example, in 2007,20 we showed that we could grow elongated PFS53−PI320 (PI: polyisoprene, subscripts refer to the mean number of repeat units) micelles by adding PFS53−PI320 unimers to a solution of PFS53−PI320 seeds dispersed in hexane, a selective solvent for the PI block. We showed that the lengths of the resulting micelles depended only on the ratio of the mass of unimers added to the solution to the mass of seeds in the solution.21 We also demonstrated that the (semicrystalline) PFS block provided the driving force for micelle growth: addition of a solution of PFS40−PDMS330 (PDMS: polydimethylsiloxane) unimers to a solution of PFS53− PI320 seeds resulted in the formation of MPFS−PDMS-b-MPFS−PI-bMPFS−PDMS triblock comicelles. This important feature of PFS block copolymers allowed us to prepare other types of triblock comicelles,22,23 even enabling us to make multiblock comicelles with alternating fluorescent blocks of controlled length that could serve as one-dimensional barcodes.24,25 Another step in the control of PFS-based micelles was recently achieved by Rupar et al.26 (Scheme 1c) who developed an elegant procedure to form asymmetric multiblock comicelles, with two or three different micelle blocks. These micelles were characterized by TEM, with Pt staining to identify the PI block. In this work, we introduce a simple approach based on the concept of “Russian dolls” to evaluate the form factor of asymmetric elongated structures composed of multiple blocks. The equation obtained from this approach is valid not only for rigid cylinders made of blocks with different cross sections (thin, circular, elliptic, and tubelike) but also for more complex structures such as elongated Janus micelles or patchy micelles. We are also able to account for the polydispersity in length of all the blocks. In this paper, we will introduce our new approach and compare the analytical results with Monte Carlo simulations and results obtained in the literature. We will also evaluate the effect of the length of the blocks as well as the asymmetry of the rod on the form factor. Finally, we will apply the form factor, in conjunction with TEM image analysis, to the

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MODEL DEVELOPMENT The scattering form factor of an object, P(q), is the Fourier transformation of its distance distribution function for an ensemble of randomly oriented particles, p(r):27 P(q) = 4π

∫0

∞

p(r )

sin(qr ) dr qr

(1)

where, r is the distance between two points in the scattering object and q is the length of the scattering vector. In the simplest case, for a thin rigid rod, p(r) decreases linearly as a function of r: p(r) = a − br, which after integration over the total length of the rod, LT, leads to the well-known form factor of a thin rigid rod: 2 P(q) = qL T

∫0

qL T

2 ⎡ 2 ⎛ qL T ⎞⎤ sin(z) ⎜ ⎟ dz − ⎢ sin ⎥ ⎝ 2 ⎠⎦ z ⎣ qL T

(2)

27

Since the scattered waves are coherent, one can also calculate the form factor of an object by first summing the amplitude of all the waves scattered by the scattering object, at each scattering vector.28 In such a case, the amplitude, F(q), of the form factor is given by F(q) =

∫0

V

ρ(r )e−iqr dV

(3)

where, for light scattering, ρ is the polarizability parameter, and V is the volume of the scattering object. For a thin rigid rod, with an angle β between q and the rod axis,29 the amplitude of the form factor is given by ⎛ qL cos(β) ⎞ F(q) = j0 ⎜ T ⎟ ⎝ ⎠ 2

(4)

where, j0 is the zero-order spherical Bessel function of the first kind. The form factor of the scattering object is then obtained by multiplying F(q) by its complex conjugate, F*(q) and averaging the product over all the possible orientations of the object (over all values of β). The form factor of a thin rigid rod obtained in this way is of course equal to the one given in eq 2. Using the pair distribution function (eq 1), Holtzer and Rice calculated the form factor of an inhomogeneous thin rigid rod12 made of three different blocks (Figure S1 of the Supporting Information). They first evaluated the pair distribution function inside each block, next, for scattering points located in two B



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adjacent blocks, and finally, when the scatterers were in the blocks located at the two extremities of the rod. The expression developed by these authors is valid for triblock thin rigid rods made of blocks of any length and allowed them to evaluate the form factor of myosin filaments made of two linear arrangements of H (heavy) and L (light) meromyosin subunits: LHL and LLH. Their equation (reproduced in Supporting Information) is long and difficult to simplify; modification of the expression to introduce different cross sections or polydispersities in the different block lengths is also not straightforward. To simplify the equation and apply it to more complex structures (thick rods with different cross sections, Janus micelles or blocks of polydisperse length), we chose to describe inhomogeneous rods using the concept of “Russian dolls” (i.e., each rod is nested within the limits of a longer rod), starting from the mother rod (the longest), next the daughter rod, and then granddaughter rod, etc. Figure 1 gives two representations

center of mass of the mother rod, and rd is a vector giving the position of the center of mass of the daughter rod. The terms e−iqrm and e−iqrd are phase factors accounting for the position of the mother rod and the daughter rod, respectively. We also want to point out that Fm(q) contains only information related to the mother rod (its length, Lm, and its cross section), while Fd(q) needs to account for the signal from the daughter rod after subtracting the signal from the section of the mother rod occupied by the daughter rod. The form factor of the rod, P(q), is then obtained by integrating the product F(q)F*(q) over all the possible orientations of the rod. By choosing a coordinate system such that the vector q is set along the z axis, and β is the angle between q and the rod axis (see Figure S2 of the Supporting Information), we can rearrange the form factor: P(q) =

1 2

∫0

π /2

(FmFm* + FdFd* + FmFd*e−iq·s + Fm*Fdeiqs)sin(β) dβ [ρm (Vm − Vd) + ρd Vd]2 (6)

where s = rm−rd, Vm is the volume of the mother rod and Vd, the volume of the daughter rod. Please note that to simplify the writing of the equation, Fm(q), F*m(q), Fd(q), and F*d(q) have been replaced by Fm, F*m, Fd and F*d, respectively, in eq 6 as well as in the following equations. In many cases (thin rod, cylinder with circular or elliptical cross sections, tube) the amplitude of the form factor is a real function and the form factor shown in eq 6 becomes Figure 1. Two representations of an asymmetric triblock thin-rigid rod. Lm is the length of the “mother” rod, Ld is the length of “daughter” rod nested within the mother rod and shifted from the center of the mother rod by a distance, s. s varies from s = 0 (symmetric triblock thin rigid rod) to s = (Lm − Ld)/2 (diblock thin rigid rod). In (A), the mother rod is a thin rigid rod and the daughter is a core/shell rod, while in (B), the mother rod is a core/shell rod and the daughter rod a thin rigid rod. Example (B) is representative of the MPFS−PI-b-MPFS−PDMS-b-MPFS−PI triblock comicelles in decane studied in the second part of this paper, since the PDMS corona is contrast matched in decane.

P(q) =

∫0

π /2

[Fm 2 + Fd 2 + 2FmFd cos(ϑ)]sin(β) dβ [ρm (Vm − Vd) + ρd Vd]2 (7)

where ϑ = qscos(β). Once the amplitude factor of the rod is known, we can easily evaluate the form factor of the Russian doll. One can, for example, develop the form factor of relatively simple structures for any kind of cross section of the mother rod, ψm, and of the daughter rod, ψd (cylindrical, elliptical, tubelike, rectangular, or core/shell), since Fm(q) and Fd(q) are given by

of such a system, with a mother rod of length Lm and its daughter of length Ld. The daughter rod has to be nested within the mother rod along the main axis of the rod, while the diameter of the daughter rod can either be larger (Figure 1A) or smaller (Figure 1B) than that of the mother rod. To fully describe this system, one needs to know the shift parameter, the variable s, that parametrizes the distance between the center of mass of the mother rod and the center of mass of the daughter rod, as well as the contrast (polarizability parameter for light scattering) of the mother rod, ρm, and daughter rod, ρd. The variable s is set as positive, and its value can only vary from s = 0 (the center of mass of both rods are located at the same position) to s = (Lm − Ld)/2, leading to the formation of a diblock rod. If s were to be larger than (Lm − Ld)/2 then the end of the daughter rod would be outside of the mother rod, which is forbidden by the concept of Russian dolls. The scattering amplitude of the complete structure can be described as27 F(q) = Fm(q)e−iq·rm + Fd(q)e−iq·rd

1 2

⎛ qL cos(β) ⎞ Fm(q) = ρm Lmψm j0⎜ m ⎟ ⎝ ⎠ 2

(8a)

⎛ qL cos(β) ⎞ Fd(q) = (ρd ψd − ρm ψm)Ld j0⎜ d ⎟ ⎝ ⎠ 2

(8b)

ψm and ψd are the amplitude factors for the cross section of the mother rod and the daughter rod, respectively. For thin rigid rods, ψm = ψd = 1, while for cylindrical rods of cross-section radius Rm (mother rod) and Rd (daughter rod), ψm = 2

J1[qR m sin(β)] qR m sin(β)

R m2

(9a)

and

(5)

ψd = 2

where Fm(q) is the amplitude for the mother rod, Fd(q) is that for the daughter rod, rm is a vector indicating the position of the C

J1[qR d sin(β)] qR d sin(β)

R d2

(9b)



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where J1(x) is the first-order Bessel function. Other cross sections such as elliptical, tubelike, rectangular, and core/shell, can also be incorporated in the model, as well as the amplitude of Janus micelles with the plane of symmetry perpendicular to the rod axis.

P(q) =

1 2

∫0

π /2 ⎡

⎢⎣ρm Lmψm j0

(

qLm cos(β) 2

Rewritting eq 7 for specific cases leads to known form factors. For example, the form factor of a symmetrical triblock rod30 can be deduced from eq 7 by centering the daughter in the mother rod (i.e., when s = rm − rd = 0):

) + (ρ ψ − ρ ψ )L j ( d d

[ρm (Vm − Vd) + ρd Vd]2

Moreover, if the length of the daughter rod is equal to half the length of the mother rod and located at one end, this structure corresponds to a rigid Janus rod where the rod consists of only two blocks of the same length but with different polarizability parameters. In such a case, one can replace Ld by Lm/2 and s = (Lm − Ld)/2 by Lm/4: P(q) =

1 V 2⎡⎣(ρm + ρd ) 2m ⎤⎦

2

∫0

qLd cos(β) 2

)⎤⎥⎦ sin(β) dβ 2

(10)

the analytical equation (dashed red trace) gave identical Holtzer-Casassa plots, validating our model. Figure 2 also demonstrates the influence of the shift parameter on the shape of the form factor of the mother/ daughter rod. In the specific case presented in Figure 2, one observes that when the daughter rod is centered within the mother rod (s = 0), two plateaux can easily be observed. Interestingly, the height of the first plateau, at low q, increases when s increases (i.e., when the daughter rod is not centered), and the first plateau becomes hard to discern when the daughter rod is located at one extremity of the mother rod [s = (Lm − Ld)/2]. According to eq 7, when the length of the mother and daughter rods are kept constant, the only part of the form factor which depends on the position of the two rods (and thus, on the asymmetry of the structure) is given by FmFd cos[qs cos(β)]. The changes observed in the Holtzer-Casassa plots are due to the displacement of the center of mass of the daughter rod away from that of the mother rod. Moreover, FmFd cos[qs cos(β)] tends to 0 when the product qs increases (see Figure S4A of the Supporting Information). Thus, at large q values, the form factors of the three different mother/ daughter rods will be equal. This result indicates, as one would expect, that the average linear aggregation number of the comicelles, (Nagg,L)av, would be the same for the three different structures. At an intermediate q range, however, the shift parameter directly influences the shape of the form factor (Figure S4B of the Supporting Information). The effect of the length of the mother rod versus the daughter rod can also be evaluated. In the known case of a system where the daughter rod is located in the center of the mother rod, one observes that increasing the length of the

π /2

sin(β) dβ

2 ⎛⎡ ⎛ qL cos(β) ⎞⎤ × ⎜⎜⎢ρm Lmψm j0⎜ m ⎟⎥ ⎝ ⎠⎦ 2 ⎝⎣ 2 ⎡ L ⎛ qL cos(β) ⎞⎤ ⎥ + ⎢(ρd − ρm )ψm m j0⎜ m ⎟ ⎠⎦ 2 ⎝ 4 ⎣

⎡ ⎛ qL cos(β) ⎞ ⎛ qLm cos(β) ⎞ + ⎢Lm2ψmρm (ρd − ρm ) j0⎜ m ⎟ j0⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 2 4 ⎣ ⎛ qL cos(β) ⎞⎤⎞ × cos⎜ m ⎟⎥⎟⎟ ⎝ ⎠⎦⎠ 4

m m

d 0

(11)

Equation 11 is equivalent to the equation developed by Kaya.13 Monte Carlo Simulations. We ascertained the validity of eq 6, by comparing the analytical expression of the form factor of a mother/daughter rigid rod with the form factor obtained via a Monte Carlo (MC) simulation (details are provided in Figure S3 of the Supporting Information).31,32 Subtle features of the form factor of a rigid rod can be better observed when one uses a Holtzer-Casassa33,34 representation, where qP(q)/π is plotted as a function of q. For example, the limiting plateau at high q in a Holtzer-Casassa plot obtained from a solution of uniform thin rigid rods gives the linear aggregation number, Nagg,L, in number of block copolymer molecules per nanometer. In Figure 2, we present the Holtzer-Casassa plots of three different mother/daughter rods with cylindrical core/shell cross sections obtained by MC simulation (blue curves) and using eq 6 (dashed red curves). For this comparison, we considered a system that we will present in the second part of this paper, where the daughter rod had a small cross section (radius = 9 nm), while the mother rod was a core/shell cylindrical rod with an inner radius of 9 nm and an outer radius of 22 nm. The mother rod was 450 nm long, and the length of the daughter rod was equal to 270 nm. The polarizability parameter of the shell of the mother rod, ρms, was equal to 0.012, while that of the core of the mother rod and of the daughter rod was ρmc = ρdc = 0.029. We tested the validity of our model by using three different values of shift parameters: s = 0, when the daughter rod is centered in the middle of the mother rod, s = (Lm − Ld)/2, where the daughter rod is located at one end of the mother rod, and finally s = (Lm − Ld)/4, which is an intermediate case. For each case, we note that the MC simulation (blue traces) and

Figure 2. Holtzer-Casassa plots [qP(q)Lm/π versus q] of a mother/ daughter system with core/shell cross-section as deduced by Monte Carlo (MC) simulations (blue curves) and as obtained from eq 6 (red dashed lines) for three different shift parameters: s = 0, s = (Lm − Ld)/2, s = (Lm − Ld)/4. The length of the mother rod is Lm = 450 nm, and the of the daughter is Ld = 270 nm. The polarizability parameter of the cylindrical core of the mother rod is the same as that of the daughter rod, ρdc = ρmc = 0.029, and the polarizability parameter of the corona of the mother rod is ρms = 0.012. D



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The form factor when the mother rod is polydisperse in length is thus given by

mother rod, while keeping the length of the daughter rod constant leads to a shift of the second plateau (located at a higher q region) to a lower q region, as shown in Figure 3. In such a

∞

PpolyLm(q) =

∫L +2s wm(x)P(q , x) dx d

(13c)

with bm(z m + 1) z m −bmL L e , zm

wm(L) =

bm = (zm + 1)/(Lw ,m) (13d)

where Lw,m is the weight-average length of the mother rod, and zm is the dispersion parameter of the mother rod. Finally, by combining eqs 12, 13a, 13b, 13c and 13d, one can write the form factor of a system of rods, where the daughter rod and the mother rod are polydisperse in length and where the shift parameter is broadly distributed around a given position, s. As we previously mentioned, the most difficult part in the evaluation of the polydispersity is to properly define the order and limit of the integrals. One first has to integrate the shift parameter from 0 to (x − y)/2, where x is the length of the mother rod and varies from 0 to ∞, while y is the length of the daughter rod, which varies from 0 to x. We want to point out that the shift parameter is always considered as a positive value. Thus, (x − y)/2 can only be positive, which is the case when the shift parameter varies from 0 to (x − y)/2 ≥ 0. For this case, the form factor of a solution of mother/daughter rods polydisperse in length and where the position of the daughter rod is also polydisperse is given by the expression:

Figure 3. Plots of qP(q)Lm/π versus q for the four daughter/mother rod systems with a 270 nm long daughter and mother rods of length 450 nm (black curve), 545 nm (dashed red), 640 nm (green), and 720 nm (dotted blue). The curves were plotted considering that the daughter and mother rods were monodisperse in length, and that the shift parameter was equal to 0 (centered daughter rod).

case, the transition, again, becomes less and less obvious as the mother rod becomes longer. In Figure 3, we used the lengths of micelles studied in the experimental part of this work. Introducing Polydispersity. A more challenging task for the determination of a mother/daughter system is to account for the shift and length polydispersity of the mother and daughter rods in the form factor. We first consider the case where the mother and daughter rods are monodisperse in length, while the location of the center of the daughter rod follows a Gaussian distribution along the main axis: (Lm − Ld)/2

Ppolyshift(q) =

∫0

s (x ) d x

∫0

(Lm − 2s)

wd(x)P(q , x) dx

(12)

wm(x)wd(y)s(z)P(q , x , y , z) dz dy dx

⎛ qLgd cos(β) ⎞ ⎟⎟ Fgd(q) = (ρgd ψgd − ρd ψd)Lgd j0⎜⎜ 2 ⎝ ⎠

(13a)

(15)

with ρgd, the contrast of the granddaughter rod, ψgd, its crosssection, and Lgd, its length. The form factor given in eq 6 becomes

Choosing a Zimm-Schultz distribution to describe the distribution in length of daughter and mother rods, we can write b(zd + 1) zd −bdL wd(L) = d L e zd

x − y /2 ≥ 0

To evaluate eq 14 numerically, one has to set the integral limits of the mother rod. This can only be done by plotting the distribution in length of the mother rod, wm(L), and set the upper limit of the integral equal to the length value at which wm goes back to 0. More Complex Structures. Up to now, we have only considered a system consisting of a mother and a daughter rod; however, our approach is also valid for more complex structures. For example, by adding one more generation (a granddaughter), one can calculate the form factor of a rod made up of 5 blocks (depending on the position of the daughter and granddaughter rods within the mother rod). The amplitude factor of the granddaughter, Fgd(q), is given by

where s(x) = exp[−(x − μ)2/2σ2]; μ is the average location of the daughter rod’s center of mass, and σ is the standard deviation. To introduce the polydispersity in length of the mother and the daughter rods, one needs to keep in mind that the daughter rod has to be nested within the mother rod (i.e., 0 < Ld < Lm − 2s). These inequalities give us the limits of the integration of the form factor when the daughter rod is polydisperse in length: PpolyLd(q) =

x

∫0 ∫0 ∫0

(14)

s(x)P(q , x) dx

(Lm − Ld)/2

∫0

∞

Ppoly(q) =

P(q) = (13b)

1 2[ρm (Vm − Vd) + ρd (Vd − Vg d) + ρg d Vg d]2 ×

where bd = (zd + 1)/Lw,d. Lw,d is the weight-average length of the daughter rod, and zd is the dispersion parameter of the daughter rod. For the polydispersity in length of the mother rod, the mother rod has to be larger than the daughter rod even if the latter is shifted from the center (i.e., Lm > Ld + 2s).

∫0

π /2

sin(β) dβ × (FmFm* + FdFd* + Fg dF g*d

+ FmFd*e−iqs1 + Fm*Fdeiqs1 + FdF g*d e−iqs2 + Fd*Fg deiqs2 + FmF g*d e−iq(s1+ s2) + Fm*Fg deiq(s1+ s2)] E

(16)



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where s1 is the shift between the center of the daughter rod and that of the mother rod and s2 is the shift between the granddaughter rod and the daughter rod. In eq 16, Fgd(q) and F*gd(q) have also been written as Fgd and F*gd to simplify the writing of eq 16. The shift between the granddaughter and the mother rod is given by s1 + s2. The generations of rods also have to be organized in such a way that the shift parameters are all positive. This system is particularly useful when one wants to study an elongated object made of three distinct blocks, such as the one shown in Scheme 1c. Moreover, the distribution in length and location of the different generations can be evaluated following the approach which led to eq 14. Here again the main difficulty is to properly set the limits of the integrals, knowing that the granddaughter rod has to be nested within the daughter rod, itself nested within the mother rod.

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EXPERIMENTAL APPLICATION To test our model, we studied by static light scattering, a solution of triblock comicelles obtained by seeded growth. Following the procedure described by Rupar et al.,26 we prepared a solution of seeds using PFS76−PDMS456 block copolymer, by heating a solution of PFS76−PDMS456 in decane (c = 0.05 mg/mL) at 100 °C for 1 h. The solution was then taken out of the oil bath and left in air to cool to room temperature. The micelles formed were sonicated twice for 10 min each time. The average length of the seeds was 270 nm (see TEM image and histogram of Figure S4, panels A and B, of the Supporting Information). Then, to 1 mL of seed solution, we added different amounts of PFS50−PI1000 unimers dissolved in THF (10 mg/mL) (namely, 4.8, 7.2, 9.5, and 11.8 μL) to the seed solutions. We prepared a series of four solutions containing triblock micelles of different lengths (samples A, B, C, and D, see Tables 1 and the Supporting Information). After addition Table 1. Summary of the Amount of PFS76−PDMS456 Seeds and PFS50−PI1000 Unimers Used to Form the Triblock Comicelle Solutions, and Values of the Average dn/dc and Average Linear Aggregation Number for Each Solution sample name

PFS76− PDMS456 seeds (mg)

PFS50−PI1000 unimers added (mg)

(dn/ dc)av (mL/g)

(Nagg,L)av (molecules nm−1)a

A B C D

0.048 0.047 0.047 0.047

0.048 0.072 0.095 0.118

0.103 0.109 0.113 0.115

2.60 2.53 2.34 2.28

a

Figure 4. TEM micrographs of PFS50−PI1000/PFS76−PDMS456/ PFS50−PI1000 triblock comicelles (A) before and (B) after staining with Karstedt’s catalyst. Insets of (A and B) show a higher magnification of the corresponding micelles.

Obtained from the fitting of the experimental data using eq 17.

excess, this catalyst seems also to interact with the PFS76− PDMS456 seeds (see inset of Figure 4B), with protrusions appearing along the seed core, which could not be observed prior to staining (inset of Figure 4A). This interaction, however, did not prevent us from differentiating the PFS50−PI1000 external regions from the PFS76−PDMS456 central region (Figure 4B). The staining of the PFS50−PI1000 shows that it added to the seeds at both ends, as previously reported.26 We verified that the staining of the micelles did not affect their final length, by measuring the length of unstained micelles of sample A. The number-average length of these micelles was in agreement with the number-average length of the stained micelles (Lm_unstained = 440 nm and Lm_stained = 450 nm). We measured the lengths of the stained micelles in three steps: We measured the length of the first PFS50−PI1000 block, next the length of the nested PFS−PDMS block, and finally the

of different amounts of PFS50−PI1000, we swirled the solutions and let them age for 1 day. We took an aliquot of each solution and diluted it to a concentration of 0.01 mg/mL for static light scattering studies. We also took one drop of each solution for TEM imaging. We finally added Karstedt’s catalyst in excess to the rest of the solutions. This catalyst is known to selectively coordinate to olefins, and can, thus, be used to stain polyisoprene.35 The stained micelles were studied by TEM after an additional day of aging. In Figure 4A, we present a TEM micrograph of micelles grown to 440 nm prior to staining. In this micrograph, the PFS76−PDMS456 seeds cannot be distinguished from the PFS50−PI1000 block grown from their extremities. The distinction between both polymers became visible after addition of Karstedt’s catalyst (Figure 4B). To be certain that all the PI chains were stained, we added a large excess of catalyst. Interestingly, when added in large F



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Figure 5. Number length−frequency of triblock comicelles prepared from the elongation of 270 nm PFS76−PDMS456 seeds with PFS50−PI1000 unimers in decane. The number-average lengths of the micelles are (A) 450, (B) 545, (C) 640, and (D) 720 nm.

length of the second PFS50−PI1000 block. Here we want to point out that since the seed micelle (the PFS−PDMS block) is nested within the final micelle, it represents “the daughter rod” in our analysis. The shift parameter was deduced by taking the absolute value of the difference between the length of the first PFS50−PI1000 block and the length of the second PFS50−PI1000 block, while the total length of a micelle (i.e., the mother rod) was obtained by adding the lengths of the three blocks. TEM image analysis was performed on 200 micelles or more, for each sample. The number length−frequency of the mother rods obtained from the TEM image analysis of each micelle solution are shown in Figure 5 (panels A−D) (the number length−frequency of the daughter rods and of the shift parameter are shown in Figures S5, S6, and S7 of the Supporting Information, respectively). Values of Lm, Ld, and s, as well as their respective standard deviation in percent are summarized in Table 2. From the histograms shown in Figure 5, one notes that as the amount of PFS50−PI1000 unimer added to the solution increased, the total length of the micelles increased. In our formalism, we say that the length of the mother rods shifted to higher values. One also observes that the mother rods are slightly polydisperse in length, especially for sample C, which exhibits a tail at large micelle lengths, as confirmed by the standard deviation of this sample which was almost 10% larger than that of the other samples. Table 2 also shows that as the amount of PFS50−PI1000 added at the end of the seeds increased, the PFS76−PDMS456 part became slightly less centered in the micelles and the distribution of the shift parameter broadened slightly. This variation was, however, not very significant (less than 5%). Figure 6 shows the Holtzer-Casassa plots (qRθ/πKcMav versus q) of the four samples prepared in decane. Rθ is the Rayleigh ratio, K, the optical constant, c the total polymer concentration, and Mav is the average molecular weight of the polymer calculated from the weight ratio of the block copolymer present in the solution [Mav = x1MPFS−PI + (1 − x1) MPFS−PDMS], where x1 is the weight ratio of PFS50−PI1000 to the total amount of polymer in solution. The average refractive index increment, (dn/dc)av, was calculated for each solution: [(dn/dc)av = x1(dn/dc)PFS−PI + (1 − x1) (dn/dc)PFS−PDMS], where (dn/dc)PFS−PI is the dn/dc of

Table 2. Summary of the Number Average Lengths of the Mother Rods, the Daughter Rod, and of the Shift Parameters and Their Respective Standard Deviations Evaluated by TEM Image Analysis after Staining sample name

Lm (nm)

σLm (%)

Ld (nm)

σLd (%)

s (nm)

σs/s (%)

A B C D

450 545 640 720

20 17 27 18

270 280 280 270

28 27 27 24

10 16 20 23

77 77 85 73

PFS50−PI1000 in decane and (dn/dc)PFS−PDMS is the dn/dc of PFS76−PDMS456 in decane. Both (dn/dc)PFS−PI and (dn/dc)PFS−PDMS were evaluated according to the Dale-Gladstone relation.36 In this relation, dn/dc of each block constituting the block copolymer is calculated according to dn/dc ≈ (np − ns)/ ρp, where np is the refractive index of the polymer constituting block, ns is the refractive index of the solvent, and ρp is the density of the polymer. The dn/dc of the each block was evaluated using a density of 1.26 g/cm3 and a refractive index of 1.68 for the PFS block,37 a density of 0.92 g/cm3 and a refractive index of 1.52 for the PI block38 and a density of 0.965 g/cm3 and a refractive index of 1.40438,39 for the PDMS block. Since the refractive index of decane is 1.409, the PDMS block was considered as contrast matched. The values of the dn/dc as well as the amount of polymer added to form the triblock comicelles are summarized in Table 1. Moreover, the fits shown in Figure 6 are given by f (q) =

qPpoly(q)Lm π

Nagg, L

(17)

where Ppoly(q) is defined in eq 14, considering a rod with a circular core−shell cross section. This equation accounts for the polydispersity of the mother and daughter rods and of the shift parameter. It also considers that the micelle has a cylindrical core with a concentric shell.40 Eq 14, applied to rods polydisperse in length with a concentric core−shell cross section, contains 15 adjustable parameters. It would be unrealistic to obtain a meaningful fit without fixing several of these parameters. From previous experiments, as well as data deduced from TEM G



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Figure 6. Holtzer-Casassa plots of qRθ/πKcMav versus q for the four triblock comicelle solutions prepared from the elongation of 270 nm long PFS76−PDMS456 seeds with PFS50−PI1000 unimers, for samples A, B, C, and D. For each plot, the plain curve corresponds to the plot of qP(q) Lm(Nagg,L)av/π.

value of (Nagg,L)av shifted the fitting curves up or down, but did not affect their shape. The choice of the polydispersities in length of the daughter and mother rod and of the shift parameter might not seem to be the most appropriate parameters to use to adjust the fit of the Holtzer-Casassa plots, since these values would seem to be accessible by TEM image analysis. TEM image analysis, however, leads to number distributions, while static light scattering is affected by the weight distribution of the micelles. When the elongated micelles are uniform (formed by a single type of block copolymer and with a uniform width), one can deduce the weight distribution of the micelles from the number distribution based on TEM image analysis. In this case, we observed that the weight distribution was narrower than the number distribution.42 However, for triblock comicelles, one cannot use the number distribution to calculate the weight distribution by TEM. In the Holtzer Casassa plots shown in Figure 6 (panels A and B), one observes the presence of two well-defined plateaux. These plateaux are indicative of a composite structure (i.e., a mother/daughter system). However, the separation between these two plateaux vanishes as the mother rod became longer and more broadly distributed (Figure 6, panels C and D). From previous work, we know that Holtzer Casassa plots obtained from the scattering of uniform rods only show one plateau (see Figure S9 of the Supporting Information).21,43,44 The presence of the daughter rod nested within the mother rod thus significantly influences the shape of the experimental plots. Despite the complexity of the system, the theoretical fits follow quite well the experimental data over the full q range for samples A and B (Figure 6, panels A and B). The fit quality decreases slightly for samples C and D (Figure 6, panels C and D). For these two samples, we suspect that their broad length distributions (Figure 5, panels C and D) strongly affect the experimental Holtzer Casassa plots.

image analysis, we can limit the number of adjustable parameters to a minimum of four. To evaluate Ppoly(q), one first needs to calculate the polarizability of each block using the Lorentz−Lorenz equation: ρi =

ni2 − ns2 3 ni2 + 2ns2 4π

(18)

where ni is the refractive index of the region i and ns the refractive index of the solvent. To evaluate the radius of the core, Rcm, and of the shell, Rsm, of the PFS50−PI1000 micelles in the mother rod, we refitted SLS data that were published by our group.21 In our previous work, the micelles were also composed of PFS50− PI1000 and were investigated both by TEM and SLS, using the form factor of a plain cylinder. The micelles were found to be 620 nm long with a corona radius of 22 nm. These 620 nm long PFS50−PI1000 micelles (i.e., in the range of the micelle lengths studied in this work) could be well-fitted, considering a core/ shell cylinder with a core radius of 9 nm and a shell radius of 22 nm (see Figure S8 of the Supporting Information).41 For the core radius of the daughter rod, we assumed that its cross section was the same as that of the mother rod. Moreover, we did not have to consider the PDMS shell of the daughter rod, since PDMS is contrast matched to decane. The values of the length of the daughter rod, of the mother rod, and of the shift parameter were evaluated by TEM image analysis. The SLS data were thus fitted by adjusting only the polydispersity in length of the daughter and mother rods and of the shift parameter and by estimating the linear aggregation number of the overall system, (Nagg,L)av. Since we used the same seed solution to prepare the triblock micelles, we treated the polydispersity index of the daughter rod to be the same for all of the fits. The polydispersity index of the mother rod was adjusted to fit each set of experimental data points. Finally, changing the H



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To adjust the height of the fits shown in Figure 6, we used different values of (Nagg,L)av, as given in Table 1. As the triblock comicelle grew longer due to the addition of more PFS50− PI1000 unimers to the seed solution, the overall linear aggregation number of these triblock comicelles decreased. This decrease indicates that the linear aggregation number of the mother rod is lower than that of the daughter rod. PFS50−PI1000 has been widely studied by our group, and we have shown that its linear aggregation number is 1.9 polymer molecules nm−1.45 We could thus evaluate the linear aggregation number of the daughter rod (i.e., of the PFS76−PDMS456 seeds), according to the following equation: [(Nagg,L)av = x1(Nagg,L)PFS−PI + (1 − x1)(Nagg,L)PFS−PDMS]. The plot of (Nagg,L)av as a function of the weight ratio of PFS50−PI1000 to the total amount of polymer in solution, x1, is shown in Figure 7. The value of (Nagg,L)PFS−PDMS

Holtzer and Rice12 (Figure S1, eq S2); description of a mother/ daughter rod system in a laboratory fixed axis (Figure S2); details about the Monte Carlo simulations (Figure S3); further insights from eq 7 (Figure S4, eq S3); histograms of the length distribution of the mother and daughter rods deduced from TEM image analysis (Figure S5−S7); Holtzer Casassa plot of a rigid rod with a core/shell cross section (Figure S8); fit of the experimental data shown in Figure 6 with the form factor of a uniform core/shell cylinder (Figure S9); amounts of polymer used to prepare the four solutions of triblock comicelles (Table S1). This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Natural Sciences Engineering Research Council of Canada for their support of this research. I.M. thanks the EU for a European Research Council (ERC) Advanced Investigator Grant. The authors would also like to thank Dr. Mariá José González Álvarez, Professor Walter Richtering, and Professor Walther Burchard for fruitful discussions.

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Figure 7. Plot of the average linear aggregation number of the triblock comicelles, (Nagg,L)av, as a function of the weight ratio of PFS50−PI1000 to the total amount of polymer in solution, x1.

(1) Pochan, D. J. Approaching Asymmetry and Versatility in Polymer Assembly. Science 2012, 337, 530−531. (2) Walther, A.; Müller, A. H. E. Janus Particles: Synthesis, SelfAssembly, Physical Properties, and Applications. Chem. Rev. 2013, 113, 5194−5261. (3) Bates, F. S.; Hillmyer, M. A.; Lodge, T. P.; Bates, C. M.; Delaney, K. T.; Fredrickson, G. H. Multiblock Polymers: Panacea or Pandora’s Box? Science 2012, 336, 434−440. (4) Li, Z.; Kesselman, E.; Talmon, Y.; Hillmyer, M. A.; Lodge, T. P. Multicompartment Micelles from ABC Miktoarm Stars in Water. Science 2004, 306, 98−101. (5) Gröschel, A. H.; Schacher, F. H.; Schmalz, H.; Borisov, O. V.; Zhulina, E. B.; Walther, A.; Müller, A. H. E. Precise Hierarchical SelfAssembly of Multicompartment Micelles. Nat. Commun. 2012, 3, 710. (6) Pochan, D. J.; Zhu, J.; Zhang, K.; Wooley, K. L.; Miesch, C.; Emrick, T. Multicompartment and Multigeometry Nanoparticle Assembly. Soft Matter 2011, 7, 2500−2506. (7) Svaneborg, C.; Pedersen, J. S. A Formalism for Scattering of Complex Composite Structures. I. Applications to Branched Structures of Asymmetric Sub-Units. J. Chem. Phys. 2012, 136, 104105. (8) Svaneborg, C.; Pedersen, J. S. A Formalism for Scattering of Complex Composite Structures. II. Distributed Reference Points. J. Chem. Phys. 2012, 136, 154907. (9) Eliçabe, G. E. Scattering of Intersecting Spherical Particles in the Rayleigh-Gans Approximation. J. Colloid Interface Sci. 2011, 357, 82− 87. (10) Aida, T.; Meijer, E. W.; Stupp, S. I. Functional Supramolecular Polymers. Science 2012, 335, 813−817. (11) Pedersen, J. S. Analysis of Small-Angle Scattering Data from Colloids and Polymer Solutions: Modeling and Least-Squares Fitting. Adv. Colloid Interface Sci. 1997, 70, 171−210. (12) Holtzer, A.; Rice, S. A. Some Particle Scattering Factors for Rods with Inhomogeneous Mass Distributions. Application to the Molecular Configuration of Myosin. J. Am. Chem. Soc. 1957, 79, 4847−4851. (13) Kaya, H. Scattering Behaviour of Janus Particles. Appl. Phys. A: Mater. Sci. Process. 2002, 74, s507−s509.

deduced from this plot is 3.2 polymer molecules nm−1. This value is similar to those obtained from PFS block copolymers with a similar block ratio (PFS48−PI264 and PFS50− PDMS285).45 The difference in linear aggregation number between the daughter rod and the mother rod also emphasizes the influence of the corona forming block on the packing of the polymer chains inside the elongated micelles.

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CONCLUSION We have developed a new approach to calculate the form factor of asymmetric one-dimensional structures. This approach, based on the principal of Russian dolls, can be applied to any kind of cross section and for any number of blocks along the main axis. The equation of the form factor also allows us to consider the polydispersity in length of each block. We showed the influence of the shift parameter and of the length of the mother rod on the rod form factor. As a practical example, we studied, by TEM and SLS, four solutions of MPFS−PI-b-MPFS−PDMS-b-MPFS−PI triblock comicelles slightly polydisperse in length. We used TEM image analysis to independently measure the lengths of the different blocks of the micelles, as well as their shift parameter. We could then successfully combine the TEM results with our model to fit the light scattering profiles of these rigid triblock comicelles. This last result not only validated our model but also showed that the combination of scattering techniques with TEM image analysis represents a powerful tool to study self-assembled structures in solution.

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REFERENCES

ASSOCIATED CONTENT

S Supporting Information *

Additional experimental details about light scattering experiments and TEM measurements (eq S1); model derived by I



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J


Form Factor of Asymmetric Elongated Micelles: Playing with Russian

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