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Negative Diamagnetic Anisotropy and Birefringence of Cellulose Nanocrystals Bruno Frka-Petesic,*,†,∥ Junji Sugiyama,‡ Satoshi Kimura,⊥ Henri Chanzy,*,† and Georg Maret§ †

Centre de Recherches sur les Macromolécules Végétales (CERMAV-CNRS), Université Grenoble Alpes, F-38000 Grenoble, France Research Institute for Sustainable Humanosphere, Kyoto University, Gokasho, Uji, Kyoto 611-0011 Japan ⊥ Department of Biomaterials Science, Graduate School of Agricultural and Life Sciences, The University of Tokyo, Tokyo 112-8657, Japan § Department of Physics, University of Konstanz, D-78457 Konstanz, Germany ‡

S Supporting Information *

ABSTRACT: We report magnetic birefringence measurements up to high fields (17.5 T) of dilute aqueous suspensions of rod-like cellulose nanocrystals with well characterized distributions of lengths, widths and thicknesses. We compare these data with three models, one with colinear (1), one with perpendicular cylindrically symmetric tensors for diamagnetic susceptibility and refractive index (2) and one with biaxial diamagnetic anisotropy (3). We find that taking into account polydispersities of length, width, and thickness is essential for accurate fitting and that model 1 is the most appropriate, presumably because of the twisting of the suspended nanocrystal along their long axis. The best-fitted susceptibility anisotropy was Δχz(xy) = χzz−(χxx+χyy)/2 = −2.44 × 10−6 when considering only the crystalline core of nanocrystals and, more appropriately, Δχz(xy) = −0.95 × 10−6 when including crystalline core and skin. The latter value is slightly higher than Δχz(xy) = −0.68(5) × 10−6 deduced from estimations using Pascal’s additivity law. The specific birefringence of the nanocrystals in water was found to be δn0 = +0.120(2), which is well accounted for by the intrinsic birefringence of crystalline cellulose (δnintr 0 = n∥−n⊥ = +0.0744) and the birefringence arising from the slender shape of nanocrystals.

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INTRODUCTION It is well established that polymer chains may be aligned by magnetic fields, and as a consequence, the use of magnets in polymer processing may lead to polymer materials with enhanced properties.1,2 Well-documented cases of magnetic field-induced alignment include polymers organized in liquid crystalline order, especially in the nematic or chiral nematic states.3−7 For such ordered systems, because of the strong coupling of molecular orientation due to liquid crystalline organization, moderate magnetic fields are sufficient to produce very strong macroscopic alignment. Liquid crystalline epoxy polymers are examples where the processing under magnetic field has led to significant improvements of physical properties.8,9 Polymer crystallization is another area where magnetic fields have been shown to induce substantial morphological modifications that could lead to a number of advanced materials.10−14 In that case, it is believed that it is the transient mesophase15−17 in the polymer melt prior to crystallization, which is responsible for the orientation by magnetic field. Another example of successful magnetic orientation of polymers and biopolymers are fibers or rod-like structures. In these cases, the enhanced susceptibility to magnetic fields results from the anisotropic three-dimensional organization of the molecules within these elongated structures. It is this © XXXX American Chemical Society

organization and its resulting cooperativity that explains why polyethylene or carbon fibers can be oriented by magnetic fields.18 It explains also why microtubules,19 fibrin structures,20,21 or pollen tubes22 can be aligned when they are grown within magnets. Organic molecules align in magnetic fields due to the anisotropy of their diamagnetic polarizability, which is proportional to their volume V and the anisotropy of their diamagnetic susceptibility tensor χ along a specific direction i. This anisotropy can be accounted for by the differences of each pair of directions Δχij = χii − χjj and is sometimes simplified as the difference Δχi(jk) = χii − (χjj + χkk)/2. When the anisotropy is only uniaxial, the anisotropic susceptibility is generally denoted Δχ and the anisotropic direction is implicit. While for small molecules in liquids or gases, Δχ and Vand hence their degree of alignment even in strong magnetic fieldsare very small, it becomes much bigger for larger molecules or for colloidal particles. As molecular diamagnetic anisotropy usually goes along with some optical anisotropy, Δχ can be measured with strong magnetic fields H by recording the value of the Received: October 6, 2015 Revised: November 30, 2015

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DOI: 10.1021/acs.macromol.5b02201 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules optical birefringence Δn induced in the sample by the field. This is done in the classical Cotton−Mouton experiment23 that can be carried out with extreme sensitivity.2,24 For polymers, the value of the Cotton−Mouton constant CM = Δn/λH2 (with λ being the wavelength used) depends on the orientational correlations between the monomer units along the polymer chains or between neighboring chains. With polymers in dilute solution, the Cotton−Mouton constant is found to be proportional to the polymer persistence length. Its value is therefore a good indication of the polymer chain rigidity in a given polymer/solvent system.2,24,25 There is a number of polymer families for which Cotton−Mouton experiments have been carried out. With biopolymers in solutions, most of the work has been done with polypeptides26 or nucleic acids.24,27 Examples for magnetic alignment of rodlike biological particles are tobacco mosaic virus28,29 and bacteriophages fd.30 The present work deals with the diamagnetic anisotropy of native cellulose nanocrystals obtained from high field Cotton− Mouton experiments. Earlier studies on the diamagnetic anisotropy of cellulose were somewhat inconclusive. In one study,31 it was shown that flax fibers, consisting of parallel cellulose crystals tended to orient parallel to a magnetic field. This observation contradicted other experiments on ramie−a sample having a crystallinity and crystalline orientation very similar to flax−where the fibers became oriented perpendicular to the magnetic field.32 Similarly, a perpendicular orientation was also found for cellulose samples from teakwood.33 In a previous 1992 report34 we demonstrated that isolated nanocrystals from tunicin (i.e., animal cellulose from tunicate), when suspended in water under dilute nonflocculating conditions, became aligned with their long axis perpendicular to a magnetic field of 7 T, indicating a negative diamagnetic anisotropy with respect to their long axis. Our report has been followed by numerous studies, dealing in particular with the magnetic orientation of more concentrated cellulose nanocrystal suspensions organized in chiral nematic order: in these, the chiral nematic axis becomes readily oriented parallel to the applied field.35,36 Such a magnetic field induced orientation has proven useful for (i) the processing of cellulose-based nanocomposites,37−43 (ii) the improvement of the iridescence in cellulose films cast from chiral nematic crystallite suspensions,44,45 (iii) the magnetic orientation of proteins dispersed in concentrated cellulose nanocrystal suspension for NMR residual dipolar coupling measurements, etc.46 Further interesting cellulose orientation and patterning were obtained by subjecting chiral nematic suspensions of cellulose nanocrystal to either rotating magnetic fields,36 dynamic elliptical field47 or inserting a field modulator within the magnet.48 Despite these numerous applications, which now confirm the negative diamagnetic anisotropy of crystalline native cellulose, the exact value of this susceptibility Δχ is not very well-known: values of −10−7 SI deduced from oscillation measurements of cellulose fibers32,49 and −10−8 SI50 have been proposed in the literature for semicrystalline cellulose fibers. In the present work, our goal was to obtain more reliable Δχ by measuring the buildup of the optical birefringence of dilute aqueous suspensions of cellulose nanocrystalsthe basic structural elements of cellulose fiberssubjected to magnetic fields of increasing strength. Because of the geometry of the cellulose nanocrystals, and the possibility for them to orient in a biaxial manner, the nature of the diamagnetic anisotropic susceptibility

was investigated using three possible models, two of which are uniaxial and the third assuming biaxial magnetic properties. From the tendency to saturation of the birefringence in high fields, we also obtain the specific birefringence δn0 of the cellulose nanocrystals in suspension, which is essential to extract Δχ from the behavior of the nanocrystals in the field. The experimentally deduced Δχ can be compared with a calculated value obtained by using Pascal’s principle of additivity of the anisotropies of the individual bonds within the cellulose crystal.

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MATERIALS AND METHODS

1. Preparation of the Nanocrystals. Nonflocculating aqueous suspensions of tunicin (i.e., animal cellulose from tunicate) nanocrystals from the mantles of Halocynthia roretzi were prepared as described earlier.51 The concentration of each specimen was determined by vacuum drying aliquots of the suspensions at 60 °C until constant weight. 2. Measuring the Geometric Parameters of the Nanocrystals from Transmission Electron Microscopy Data. Whereas the measurement of the lengths of the nanocrystals and their distribution is relatively straightforward, the estimation of their widths and thicknesses and the correlations of these measurements with those of their lengths are hampered by a substantial uncertainty. 2.1. Lengths Measurements. Drops of nanocrystal suspensions were deposited on carbon coated electron microscope grids and allowed to dry. Images of the nanocrystals were recorded with a Philips EM400T at an acceleration voltage of 120 kV under low dose illumination conditions. Statistical length (La) distributions of the nanocrystals were obtained by measuring the length of a large number of nanocrystals in the images. The low dose conditions did not allow us to record precise widths measurements of the nanocrystals. 2.2. Widths and Thicknesses Measurements. In one method, the innermost layers of as-received H. roretzi mantles were embedded and cross-sectioned as described elsewere.52 TEM diffraction contrast images showing a distribution of cellulose crystalline microfibril cross sections were recorded in bright field mode53 with a JEOL 2000 EXII operated at 200 kV under low electron dose conditions. Under these conditions, only the crystalline areas where the microfibril axis is parallel to the electron beam display a strong black contrast, resulting from the elimination of the diffracted beams by the objective aperture diaphragm. As the core of each microfibril corresponds to a cellulose monocrystalline domain, these images led to the recording of the distribution of the widths and thicknesses of these microfibril crystalline core. Another method reported by Elazzouzi-Hafraoui et al.,54 based on the observation of negatively stained nanocrystals, gave the values of their lengths and widths in a precise way, but their thickness measurements, which were recorded at areas where the nanocrystals were sitting edge-on, due to their inherent twisting, were less precise. In the negatively stained images, the measured widths of the nanocrystals were larger than those obtained in the previous method since in addition to the crystalline core, they also included the noncrystalline surface skin of the nanorods. 2.3. Polydispersity of the Samples. Typical electron micrographs of the samples used in this study are shown in Figure 1, which corresponds to a preparation of tunicin nanocrystals. They occur as nonaggregated slender straight rods of various lengths, but of width in the range of 18 to 25 nm. As described elsewhere34,51 the core of each rod is in fact a perfect single nanocrystal of cellulose Iβ, with the chain axis of cellulose aligned along the rod axis. A distribution of the crystal lengths (La) is given in Figure 2, which shows that this distribution is skewed: while most of the nanocrystals have lengths below 1 μm, a few are longer than 10 μm. The maximum of their population occurs around 500 nm while their median length (La0) is 863 nm and the average length ⟨La⟩ is 1186 nm. The probability density of La is well described by a log-normal function PLN (La) defined as B


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Figure 1. Low dose electron micrograph of a typical aqueous dispersion of tunicin nanocrystals.

Figure 3. Low dose electron micrograph taken in bright field diffraction contrast of a cross section of the innermost layer of the mantle of a H. roretzi specimen. When the individual microfibrils are parallel to the electron beam, their crystalline cores are seen as dark spots.

Figure 4. (a) as in Figure 3, but at higher magnification. (b) same as part a, but after contrast enhancement. (c) Contour ellipses drawn around each microfibril section. (d) Typical ellipse, with the two main axes Lb and Lc. correspond respectively to the aspect ratio and the area of each section trace (Figure 4, parts c and d). This allows us to determine the relevant width Lb = π1/2a and thickness Lc = π1/2b of each crystalline core and analyze their distribution function. The distributions of Lb and Lc are presented in Figure 5a, whereas the correlation between Lb and Lc is shown in 5b. It may be noticed in this figure that the Lb and Lc are normally distributed and have a strong correlation. This can be accounted for by using a bivariate normal distribution Pbc(Lb,Lc, ρbc) (see Supporting Information, sections 1 and 2), with the corresponding correlation ρbc between Lb and Lc defined as

Figure 2. Length distribution P(La) of the tunicin nanocrystals: (a) as determined experimentally from image analysis and (b) as constructed using the fitting parameters in part a, in order to compute the polydisperse case (as a projection of Pabc(La,Lb,Lc) on the axis La).

PLN(La , La0 , σa) dLa =

⎛ ln 2(L /L ) ⎞ 1 a a0 ⎟ dLa exp⎜− 2π σaLa 2σa 2 ⎝ ⎠

ρbc =

cov(Lb , Lc ) 1 = σbσc N

N

∑ i=1

(Lb − Lb0)(Lc − Lc 0) σbσc

(2)

Here (Lb0, Lc0) and (σb, σc) are the unbiased estimators of the mean value and the standard deviation of Lb and Lc respectively. Regarding the data recorded by Elazzouzi-Hafraoui et al.,54 on negatively stained tunicin nanocrystals, an analysis of their distribution allowed these authors to establish a marked correlation ρab between the length La and the width Lb, both being in this case described by log-normal distributions. By using their raw distribution, we managed to extract a value of the correlation ρab between ln(La) and Lb, defined as cov(ln(La),Lb) /(σa σb) = 0.587, while the correlation between ln(La) and ln(Lb), could be defined as cov(ln(La), ln Lb) /(σa σb) = 0.442. One can notice that their average length ⟨La⟩ = 1163 nm is

(1)

A cross section of a fragment of tunicate mantle is shown in Figure 3. In this image, taken in bright field, under diffraction contrast mode52,53 the traces of the crystalline sections of the individual cellulose microfibrils are seen as dark spots. At higher magnification these traces appear elongated (Figure 4a). Their actual shape is revealed after contrast increase (Figure 4b). With the help of an ImageJ built-in algorithm, ellipses are drawn with major and minor semiaxes a and b, so that the aspect ratio a/b = Lb/Lc and the area A = πab = LbLc C


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Figure 5. (a) Distribution of the widths (Lb in blue) and thicknesses (Lc in red) of the tunicin microfibril sections. (b) Experimental sampling of 212 rods illustrating the correlation between Lb and Lc. (c) Projection of the trivariate distribution Pabc(La,Lb,Lc) on the (Lb,Lc) plane (in gray scale) showing the correlation between Lb and Lc. (corresponding to the size distribution Distrib1 obtained from our size distribution analysis). Projections on different planes are available in the Supporting Information for both distributions.

Table 1. Size Distribution Parameters of Tunicin Nanocrystalsa Distrib

type

La0

⟨La⟩

σa b

Lb0

⟨Lb⟩

σbc

Lc0 = ⟨Lc⟩

σc

ρab

ρbc

ρac

1 2

LNN LLN

863 932

1186 1163

0.80 0.67

12.8 21.9

12.80 24.05

2.07 0.46

7.70 9.20

1.90 0.23

0.587b 0.442

0.620 0.650c

0 or ρab 0 or ρbc

a

The distribution type refers to log-normal (L) and normal (N) distributions for (La,Lb,Lc). Distrib1 refers to a distribution where the crystals were cross-sectioned and observed by TEM in diffraction contrast mode. Distrib2 refers to a distribution where the crystals were observed by TEM after negative staining.54 The Li values are in nm, σa is the standard deviation of ln(La), σb is the standard deviation of Lb (Distrib1) and ln(Lb) (Distrib2). As no values were available for ρac, we replaced it by zero or either ρab (LNN) or ρbc (LLN) according to the distribution type. bEstimated from Distrib2. cEstimated from Distrib1 and Lc, two possibilities were investigated, the first assuming no correlation at all (ρac = 0) and the second one assuming the same value as for other comparable correlations: (Distrib1, ρac = ρab; Distrib2, ρac = ρbc). 3. Magnetic Birefringence: Cotton−Mouton Experiments. The magnetic birefringence measurements were carried out at the Grenoble High Magnetic Field Laboratory, MPI-CNRS. Figure 6 shows a schematic sketch of the experimental setup used throughout this work. This setup is similar to the one described earlier2 except that a water-cooled 19 T horizontal bore split-coil Bitter solenoid was used. In this magnet, the axial bore had a diameter of 5 cm. Two 4 mm radial horizontal bores orthogonal to the field direction allowed optical access for the beam of a 10 mW He−Ne laser perpendicular to the field direction. The magnetic birefringence Δn = n∥ − n⊥, with n∥ and n⊥ being the refractive indices parallel and perpendicular to the field, respectively, was measured with a sensitivity of the order of 10−10 at a wavelength of λ = 632.8 nm. Magnetic birefringence data were accumulated in magnetic fields H ranging from 0 to 140,000 kA/m (corresponding to field inductions B = μ0 H ranging from 0 to 17.5 T), using dilute suspensions of tunicin nanocrystals in a concentration range between 0.03 and 0.1 g/L. The suspensions were poured in quartz cells with a 1 cm path length, which were positioned inside the center of the magnet bore, within a

slightly different from ours, but correspond to samples that were produced under slightly different conditions. Moreover, their ⟨Lb⟩ is substantially higher than the one deduced from the diffraction contrast images in the cross sections. This difference likely originates from the fact that only the diffracting part of the microfibrils was considered in the diffraction contrast images, whereas in negatively stained specimens the whole width of the nanorods, including the surface cellulose chains, was considered. The discrepancy may also result from the lack of resolution in the cross section images, which were recorded under limited electron dose, as opposed to the negatively stained images, which do not need such limitation and are therefore more precise. For the sake of comparative study, we decided to combine our different sources of information regarding the size distribution of tunicin crystals and to use two sets of size distribution parameters summarized in Table 1. In particular, we assumed the missing correlation parameters in each distribution, from the other one, when available.54 In order to account for a general description of the polydispersity of the crystals in all three dimensions La, Lb and Lc, including all mutual correlations (as illustrated on Figure 5c), we used a general trivariate distribution Pabc (La,Lb,Lc), described in details in the Supporting Information (cf. Sections 1 and 2, eqs 1-6). Finally, because we did not have access to the correlation ρac between ln(La) D


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Figure 6. Schematic of the experimental setup used for the magnetic birefringence measurements. temperature stabilized sample holder kept at a temperature of 20 ± 0.1 °C. Up and down sweeping of the field was controlled at different sweeping rates of the current in the magnet solenoid ranging from 150 A/s to 30 A/s. Under these conditions, a full sweep from 0 to 17.5 T and back to 0 T took from around 6 to 30 min. Some high precision experiments were also carried out with a sweeping rate of 5 A/s. In the latter experiments, only the low field parts of the birefringence curves were measured. In all experiments, the sweeping rate was adapted to the specimen concentration (viz. viscosity) such that no time lag in the birefringence values between the up and down scans of the field could be observed. Thus, the Δn vs B curves correspond to orientation distributions of the nanocrystals in thermal equilibrium.

volume fraction Φ = c/ρ of cellulose crystals, being here Φ = 2.25 × 10−5. At high field, Δn/Φ shows a clear tendency toward saturation near −0.06, indicating that a nearly complete orientation of the crystals took place in the highest fields available. As found earlier,34 the magnetic birefringence of these suspensions was negative with respect to the direction of the applied field, and since in cellulose the refractive index is highest in the chain direction55 this unambiguously confirms that the tunicin nanocrystals tend to orient perpendicular to the field. Similar experiments were also performed with suspensions of concentration 0.110 g/L. Since they gave birefringence data nearly indistinguishable from those shown in Figure 7 despite extensive change in the concentrations in the corresponding samples, we conclude that the experiments were done under conditions where interparticle interactions were essentially negligible. The absence of interparticular interactions can also be anticipated from the comparison between the value of Φ = 2.25 × 10−5 of our measurements with the overlapping volume fraction Φ*. The latter can be calculated taking into account the volume of a nanocrystal, its sphere of influence and the maximal packing factor of these spheres, which yields

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RESULTS 1. Cotton−Mouton Experiments. Figure 7 shows typical birefringence data induced by the field on the suspensions of tunicin nanocrystals of concentration c = 0.0367 g/L. As the density of cellulose tunicin is ρ = 1630 kg/m3, one can normalize the experimental birefringence by the corresponding

Φ* = La̅ L̅bLc̅ ×

6 π × = 3 3 2 πLa̅

2 L̅bLc̅ La̅

2

= 9.0 × 10−5 (3)

As Φ is about four times smaller than Φ*, interparticular interactions can be neglected. 2. Analysis of the Cotton−Mouton Experiments. The field induced birefringence of a suspensions of birefringent particles in an electric or magnetic fields (called the Kerr effect or Cotton−Mouton effect, respectively) has been investigated in a variety of cases in terms of permanent and induced dipoles in order to extract the corresponding permanent polarizations (respectively magnetization) and electric (respectively magnetic) polarizabilities of colloidal particles along specific

Figure 7. Cotton−Mouton measurements on a c = 0.0367 g/L aqueous suspension of tunicin nanocrystals, current sweep rate 150 A/ s. E


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Macromolecules directions, and therefore their electric56 (respectively magnetic) susceptibilities. Analytical solutions for the induced birefringence have been derived when both permanent and induced dipoles are present on a particle with cylindrical symmetry,57 while Shah58 extended the case to noncylindrical particles with a permanent dipole oriented perpendicular to the largest induced dipole susceptibility. A more general expression has been proposed by Holcomb and Tinoco59 for any particle with no restrictions for its permanent and induced dipole moments along the three directions. However, the approach of these authors does not provide a true analytical solution as it is based on a twodimensional infinite Taylor expansion of the field intensity, and as such does diverge at finite field intensity even for a Taylor expansion reaching computing limits. One can note that the case derived in the work of Holcomb and Tinoco59 assumes that the orientational eigen-axes are identical to the optical (birefringent) eigen-axes of the particle, which is actually not the case in cellulose crystals, as we will explain later. As cellulose is a diamagnetic material, there is no permanent dipole (no permanent magnetization) and thus, only induced dipoles will be considered hereafter. In order to extract the diamagnetic susceptibility of cellulose crystals from the experimental Cotton−Mouton experiments, we explore the possible birefringence curves expected for crystals of known dimensions, for which we adjust the diamagnetic susceptibility so as to fit the experimental data. As aforementioned, we could demonstrate that the effect of the field was to align the long axis of the cellulose crystals perpendicular to the field.34 In addition, as seen below in the section 3 of this report, the field is believed to also induce an extra rotation of the nanocrystals about their long axis, leading to align the mean plane of the anhydroglucose unit (AGU) ring of the cellulose monomers perpendicular to the field. In order to account for the experimental data, we tested three models to express the anisotropy of the diamagnetic susceptibility of the crystals. In model 1, the diamagnetic susceptibility Δχ is negative and uniaxial along La, i.e. the long axis tends to orient perpendicular to the magnetic field H and no further orientation with respect to (Lb, Lc) occurs. In model 2, the diamagnetic susceptibility Δχ is positive and uniaxial along an axis contained in the plane (Lb, Lc), thus perpendicular to La, i.e., another axis than La tends to align parallel to the magnetic field H and therefore lays La perpendicular to H. In model 3, the diamagnetic susceptibility Δχ is biaxial and combines both effects described in model 1 and model 2. As such, this last model appears to be a generalized form of the previous two models, and might be considered sufficient. However, the lack of analytical solution for this model makes this model more difficult to compute. In section 2.1, we describe the three models and their theoretical expressions for a given geometry of cellulose crystal, justified initially by symmetry considerations. In section 2.2, we consider the best fit of the birefringence assuming an assembly of monodisperse rod-like crystals with average dimensions identical to those established through size distribution analysis, for each of the three considered models. In section 2.3, we introduce the distributions and correlations between the established dimensions of the crystals and consider the corresponding optimal fit of Δχ through a least-squares fitting method for the three considered models. 2.1. Modeling the Birefringence of the Cellulose Nanocrystal Suspension under Magnetic Field. We consider a

suspension of noninteracting slender parallelepiped rods of dimensions La ≫ Lb > Lc. Each rod is made of a single cellulose nanocrystal whose diamagnetic properties cause the orientation of the rods under strong magnetic field, which in turn modifies the optical properties of the suspension and causes birefringence. Before describing each model, we consider the symmetry properties of the crystals regarding diamagnetic and dielectric optical properties to justify the description used further in this work. 2.1.1. Symmetry Considerations Regarding the Diamagnetic Properties. From the crystallographic structure of the cellulose, discussed in more depth in section 3 (and in Supporting Information, section 4), we know that within the crystal, the cellulose polymer chain axis, denoted z in Figure 10, corresponds to the La direction of the rods, while the direction perpendicular to the cellulose glucosidic rings, denoted x in Figure 10, is roughly at 45 deg with respect to the Lb and Lc directions given by the parallelepipedic outer shape of the crystals. For this reason, we will hereafter refer to this direction as the b+c direction, while the third axis, denoted y, will be referred to as b−c. One can note that the magnetization of cellulose under strong magnetic field is negligible (usually 5 orders of magnitude smaller than the applied magnetic field, as it is usually the case for diamagnetic materials), and therefore no correction of the intrinsic diamagnetic susceptibility of cellulose is needed to account for the aspect ratio of the crystals, although the latter is large. Coming back to the three models, model 1 assumes a χa lower than in the two other directions, which forces La to lay in the plane perpendicular to the magnetic field (however this model cannot account for the alignment of the AGU rings with respect to the field); model 2 assumes a χb+c higher than in the two other directions, which forces La and Lb−c to lay perpendicular to the magnetic field, and model 3 assumes both a lower χa and a χb+c higher than χb−c, forcing both La perpendicular to the field and Lb+c parallel to the field. 2.1.2. Symmetry Considerations Regarding the Dielectric Properties. As the cellulose crystals orient under the magnetic field, their anisotropic optical properties confer birefringence to the whole suspension. The birefringence of the individual crystals arises from the anisotropic dielectric susceptibility of the cellulose crystals in water at optical frequencies. Unlike the diamagnetic susceptibility, the intrinsic dielectric susceptibility of usual dielectrics is close to 1, and thus when the optical electric field polarizes the crystal, the resulting polarization produces a depolarization field that is function of the crystal shape and that is not negligible compared to the externally applied electric field. As a consequence, this depolarization field modifies the effective electric field value, and because it is geometrically dependent, the effective (extrinsic) birefringence of the crystals is strongly affected by both: the aspect ratio of the crystals and the intrinsic optical indices of cellulose and the surrounding solvent. However, only optical indices of cellulose crystals parallel and perpendicular to their long dimension are reported in literature, while to our knowledge no specific optical indices have been reported in the (Lb, Lc) plane. Therefore, we assumed in the proposed modeling the same intrinsic optical indices along Lb and Lc axes. Moreover, taking advantage of the aforementioned symmetries of the magnetic main axes, we can see that all the three previously mentioned models tend to orient the direction La perpendicularly to the applied H field, while both Lb and Lc will F


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Macromolecules equally get aligned parallel to the H field direction, no matter whether only Lb+c or both Lb+c and Lb−c are parallel to H. This means that the specific birefringence δn0 of cellulose nanocrystals has to be taken for all three cases as the difference between La and (Lb, Lc): n + nc δn0 = na − b (4) 2 2.1.3. Orientation Modeling of the Rods under Magnetic Field. The Zeeman energy of a diamagnetic object placed in a magnetic field H can be written as U=−

μ0 V 2

H ·χ ·H

γij = −

where χ is the susceptibility tensor of the crystal, V its volume, H is the magnetic field vector and μ0 = 4π 10−7 is the vacuum magnetic susceptibility in SI units. As this energy depends on the object orientation, the minimization of its energy will result in the object orientation with respect to the magnetic field H. The orientation of a single crystal of dimension (La, Lb, Lc) can be expressed in terms of three Eulerian angles, θ, ψ, and ϕ, which are measured with respect to an external Cartesian coordinate system, one axis of which is taken as the direction of the magnetic field. It is convenient to define the Euler angles accordingly to the model we need, in order to simplify the calculation. We choose the colatitude angle θ to be between the magnetic field vector and the long axis La = z of the rods. This choice is motivated by the symmetry argument showing that axes x and y have equivalent average optical properties and as such, only the orientation of the axis z (i.e., La) contributes to the birefringence. The angle ψ measures the rotation about the axis z, and ϕ is the longitude of the axis z. Eq 5 can be explicitly written in terms of Euler angles as

... −

3 [ny − nx] 4

f (θ , ψ ) =

2π

∫0 ∫0

π

2π

∫0 ∫0

sin θ dθ dψ

2π

∫0 ∫0

π

e−U / kBT 2π

π

∫0 ∫0 e−U / kBT sin θ dθ dψ

(8)

Δn(H ) = ⟨n ⟩ − ⟨n⊥⟩ 2π

∫0 ∫0

=

π

[ e ⃗ . n . e ⃗ − e⊥⃗ . n . e⊥⃗ ] f (θ , ψ )...

sin θ dθ dψ (9)

where the brackets denotes the average over all directions, e∥⃗ and e⊥⃗ are the unit vectors parallel and perpendicular to the applied field H, and n is the optical index tensor of the particles. After projecting the expression of n on the e∥ and e⊥ axes, one gets: π

⎛ 3 cos2 θ − 1 ⎞ ⎟ f (θ , ψ ) sin θ dθ dψ ⎜ 2 ⎠ ⎝

(1 − cos2 θ)(2 cos 2 ψ − 1) f (θ , ψ ) sin θ dθ dψ

Using our symmetry arguments, one can see that the second integral vanishes (ny − nx = 0) as both nx and ny have the same average values nx = ny = (nb+nc)/2. The formula thus simplifies to Δn(H ) = δn0 Φ

(7)

where the integration on ϕ has been dropped as it does not change the distribution. According to the Holcomb and Tinoco59 representation, the birefringence can therefore be calculated by the following integral:

(6)

nx + ny ⎤ ⎡ Δn(H ) = ⎢nz − ⎥ ⎣ 2 ⎦

2kBT

where the angular-independent terms of U/kBT have been dropped as they do not contribute to the crystal orientation. Here kBT stands for the thermal energy, γij is the Zeeman energy of each crystal expressed in kBT units, and associated with the anisotropic susceptibility Δχij = χii − χjj, i.e. the difference of the diamagnetic susceptibility between any principal axes i and j (i.e., x,y,z). One can note that the energy U does not depend on the longitude ϕ. According to this definition, the steady-state orientation distribution function f(θ,ψ) at equilibrium can be written as

(5)

U = γyx sin 2 θ cos2 ψ − γzy cos2 θ kBT

μ0 LaLbLc Δχij H2

(10)

integral corresponds to the nematic order parameter of the rods, usually denoted S2 provided that θ remains defined as the angle between the particle anisotropic axis and the external uniaxial direction. From a statistical point of view, the order parameter describes how particles are oriented, no matter what is their apparent shape. In the case of particles with uniaxial magnetic and optical properties, the simple case when the two axes are identical allows a straightforward definition of both the order parameter and the optical properties of the system under external field. This is the case only in one of the model that we have investigated (model 1). For the case when the two anisotropies directions are orthogonal (model 2), the magnetic anisotropic axis is relevant to derive the calculation, usually leading to different notation conventions from ours, where θ would correspond to the angle between the x axis and the external field with an adapted birefringence saturation value. Finally, when the magnetic properties are biaxial (model 3), the

⎛ 3cos2 θ − 1 ⎞ ⎜ ⎟ f (θ , ψ )... 2 ⎝ ⎠ (11)

For uniaxial systems where the angle ψ is not relevant, this double integral becomes a simple integral and the orientation distribution function is simply denoted f(θ). This integral is then referred to as the second moment of f(θ) as it corresponds to the average value of P2(cos θ) = (3 cos2 θ− 1)/2, the second Legendre polynomial of cos θ (i.e., P0 = 1, P1(X) = X, P2(X) = (3X2−1)/2, etc.). When an assembly of rod-like particles aligns uniaxially their anisotropic axis, either parallel or perpendicular to a given direction as in a nematic liquid crystal, then this G


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Macromolecules ⎛ ⎞ 3 ⎜ 4γyx e γyx 1 Σ 2(γyx) = − 1⎟ − ⎜ ⎟ 4γyx ⎝ π erfi( γyx ) 2 ⎠

use of the most general equation is required. In any of those cases, it is always possible to extend the usual definition of S2 as a descriptive quantity for the quadrupolar order parameter associated with the orientation of the axis La. S2 =

3 cos 2 θ − 1 2

(γyx ≥ 0 as Δχyx ≤ 0)

where erfi(x) = −i erf(ix) is the imaginary error function. This model corresponds to the case of a uniaxial treatment of cellulose, considering a positive Δχx(yz) along the Lb+c = x axis:

(12)

Here ⟨ ⟩ describes the average over the field dependent orientation distribution function of the rods. The parameter S2 is zero for an isotropic distribution of rod orientations (⟨cos2 θ⟩ = 1/3) at H = 0, the value S2 = +1 (⟨cos2 θ⟩ = 1) corresponds to a parallel orientation of the rods to the field, while S2 = −1/2 (⟨cos2 θ⟩ = 0) corresponds to a perpendicular orientation of the rods with respect to the field, i.e. the rods long axes are equally distributed in the plane perpendicular to the field. Using this definition, the birefringence of the suspension under magnetic field H can be rewritten as Δn fit = δn0 × S2 Φ

Δχx(yz) = χxx − (χyy + χzz )/2 = −Δχyx ≥ 0

Interestingly, the field induced nematic order parameter in this case is Σ∥2 > 0, as symmetry-wise the x-axis tends to align parallel to the field. For model 3, the diamagnetic susceptibility is biaxial, thus no unique anisotropy is defined. Instead, a tensorial description of Δχ is required, accounting for a higher susceptibility along the Lb+c = x dimension and a lower one along La = z direction: Δχzy ≤ 0 and Δχyx ≤ 0

(13)

In order to illustrate the effect of these two diamagnetic anisotropies, we introduce the ratio r as follows:

where δn0 is the specific birefringence of the rods associated with the axis La compared to the two other axes. As we shall consider in this paper only the latter case, with saturation of S2 at high fields at −1/2, one can define the saturation birefringence Δnsat as the birefringence reached at infinite H fields: Δnsat = −

r = γyx /γzy

Σ 2mix (γzy , γyx) = 1 − 3T2/T1

δn0 Φ 2

(14)

1 S2 = − Σ 2 2

∞

(15)

Γ1 =

Γi

( −γyx) p (γzy)q p ! q!

P1 = 1(p = 0);

Δχyx = 0

The corresponding expression of Σ⊥2 depends only on γzy and is given by58,60

P3 =

⎛ ⎞ 3 ⎜ 4γzy e−γzy − 1⎟ ⎟ 2γzy ⎜⎝ π erf( γzy ) ⎠

(γzy ≥ 0 as Δχzy ≤ 0)

∑ p,q=0

The mathematical expression of the function Σ2 depends on the chosen model. For model 1, the diamagnetic susceptibility along the La = z direction is lower than in the two other directions:

Σ⊥2 (γzy) = 1 +

(19)

obtained from eqs 12 and 15 where the ratio T 2 /T 1 corresponds to the calculation of ⟨cos2θ⟩ using the following Taylor expansion, as developed by Holcomb and Tinoco:59 Ti =

and

(18)

Adapting from the notations presented in the work of Holcomb and Tinoco,59 we define:

For the same reason, we find useful to introduce the function Σ2 that saturates at +1 to express the field dependence of S2 with the magnetic field H:

Δχzy ≤ 0

(17)

PP 1 3 2q + 1 P1 =

(2p − 1)! 2p − 1

2

(p − 1) ! p!

(p ≥ 1)

22pp ! (p + q)! (2q + 1)! [2(p + q) + 1]! q!

Γ2 = Γ1

2q + 1 2(p + q) + 3

(20)

As this function is based on a polynomial expansion of γzy and γyx, both quadratic in H, an infinite sum is required to avoid the function to diverge at infinite H. Moreover, the computation of this function requires the calculation of each (p,q) term of the two sums T1 and T2, which is very time-consuming when computing the polydisperse case. In practice, we overcome these two limitations using a fast computation of this function in terms of a linear combination of the two previous functions Σ⊥2 and Σ∥2 , as described in the Supporting Information, section 3, so the final function was computed as fast as the two previous models and the divergence at higher H field was suppressed. Finally, even though each crystal is biaxial, they all orient on average uniaxially as the field symmetry is uniaxial. Therefore, a nematic order parameter can still be defined in this third model and

(16)

One can note that this model corresponds to the case of a uniaxial treatment of cellulose nanocrystals, considering a negative Δχz(xy) along the La = z dimension: Δχz(xy) = χzz − (χxx + χyy )/2 = Δχzy ≤ 0

The nematic order parameter is in this case: − 2 Σ⊥2 < 0. For model 2, the diamagnetic susceptibility along the Lb+c = x axis is higher than in the two other directions: Δχzy = 0 and Δχyx ≤ 0

The corresponding expression of Σ∥2 depends only on γyx and is given by57,60 H


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Macromolecules would be equal to Σmix 2 > 0, as here again, symmetry-wise the x axis should be considered as the nematic director as saturation is reached. Following the procedure derived for the magnetic orientation of colloidal goethite nanorods,60 the specific birefringence of the rods δn0 can be expressed as

εa = n = 1.6180 εb = n⊥ = 1.5436 εc = n⊥ = 1.5436 εs = ns = 1.3314

εa − εs εb − εs n⎡ 1⎛ δn0 = s ⎢ − ⎜ ⎢ 2 ⎣ εs + Na(εa − εs) 2 ⎝ εs + Nb(εb − εs) +

⎞⎤ εc − εs ⎟⎥ εs + Nc(εc − εs) ⎠⎥⎦

The depolarization coefficients (Na, Nb, Nc) used in eq 21 are calculated from the average dimensions of the nanocrystals by circular permutation in (c → a → b → c) following a method adapted from that of Aharoni for rectangular ferromagnetism prisms.61 Here, we have considered that the elliptic sections of the nanocrystals, with axes Lb and Lc (Figure 4) are not too different from those of a parallelepiped with widths and thicknesses Lb and Lc, as these formula are mainly sensitive to the aspect ratio of the nanocrystals. In the case of Nc, the depolarization is then written as

(21)

Here, εa, εb, εc, εs are the dielectric permittivities of the cellulose in the (a, b, c) directions and of the solvent (water), respectively, at light wavelength λ = 632.8 nm and ns = εs is the optical refraction index of the solvent. According to the above arguments for uniaxiality and because only the average of εb and εc values is known, we have used the same values for the b and c directions55 Λabc =

La 2 + Lb 2 + Lc 2

Λab =

La 2 + Lb 2

Λbc =

Lb 2 + Lc 2

Λac =

La 2 + Lc 2

Nc =

(22)

2 2 ⎛ Λ − La ⎞ L 2 − Lc 2 ⎛ Λabc − Lb ⎞ 1 ⎡ Lb − Lc ⎢ ln⎜ abc ln⎜ ⎟+ a ⎟... π ⎢⎣ 2LbLc 2LaLc ⎝ Λabc + La ⎠ ⎝ Λabc + Lb ⎠

+

⎛ Λ − Lb ⎞ Lb ⎛ Λab + La ⎞ L ⎛ Λ + Lb ⎞ L ln⎜ ⎟ + a ln⎜ ab ⎟ + c ln⎜ bc ⎟... 2Lc ⎝ Λab − La ⎠ 2Lc ⎝ Λab − Lb ⎠ 2La ⎝ Λbc + Lb ⎠

+

⎛ LL ⎞ Lc ⎛ Λac − La ⎞ L 3 + Lb 3 − 2Lc 3 ln⎜ ... ⎟ + 2 arctan⎜ a b ⎟ + a 2Lb ⎝ Λac + La ⎠ 3LaLbLc ⎝ Lc Λabc ⎠

+

(La 2 + Lb 2 − 2Lc 2)Λabc L Λ 3 + Λbc 3 + Λac 3 ⎤ ⎥ + c (Λac + Λbc) − ab 3LaLbLc 3LaLbLc LaLb ⎦

2.2. Fitting the Birefringence Data Assuming a Monodisperse Size Distribution. Before incorporating the polydispersity into our models, we first analyzed the experimental Cotton−Mouton experiment assuming the three aforementioned models using a simple monodisperse case, assuming an average size for our cellulose crystals corresponding to the average dimensions derived from the distribution analysis. The experimental birefringence curve is fitted for each of the three models aforementioned using two fitting parameters, αfit and Δχij: Δn fit(H ) δn = −αfit 0 Σ 2model(γij , γjk) Φ 2

(23)

Δχij term is implicit and contained into the γij, and is, at least for model 1 and model 2, the only abscissa fitting parameter. As model 3 implies two fitting parameters in the abscissa scale either taken as (Δχzy, Δχyx) or as (Δχzy, r) - we explored two fitting conditions by fixing the r parameter to either r = 1 or r = 0.727 while keeping only one fitting parameter free for adjustment. While the choice of r = 1 is arbitrary, the second choice for r is motivated by the value expected from the analysis of the cellulose crystalline structure developed below in section 3. Finally, one can also note that model 1 corresponds to r = 0 and model 2 to r = ∞. Figure 8 reports the experimental birefringence curve with the fit for each of the 3 models, obtained by adjusting αfit and Δχij, and minimizing the difference between the calculated and experimental curves by least-squares fitting method expressed by the parameter Qfit, which corresponds to the sum of its residuals (the smaller Qfit, the better the fit, cf. Table 2). Interestingly, in the case of the Distrib1 the best fit is obtained −6 for model 1 with αmono = 0.93 and Δχmono (Qfit fit a(bc) = −5.85 × 10

(24)

Δnexp sat /[(−δn0/2)

Here, the parameter αfit = Φ] ≈ 1 is a corrective factor connecting the apparent saturation birefringence Δnexp sat and the calculated one, accounting for possible error either in theoretical determination of δn0 or in the experimental measurements of the volume fraction Φ of cellulose crystals, and is the only ordinate fitting parameter. The I


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Macromolecules

the shape of the crystals and the solvent optical index, in addition to the intrinsic optical indices of cellulose, to model their effective birefringence. 2.3. Fitting in the Polydisperse Case. In an attempt to overcome the misfit between the observed and calculated data (Figure 8) and in order to illustrate the effects of polydispersity in the data analysis, we incorporated the sample polydispersity into the model. For a polydisperse suspension of noninteracting nanocrystals, one can calculate the magnetic birefringence by integrating over the volume-weighed polydispersity distribution, as the birefringence of any subpopulation of rods in suspension is proportional to their specific birefringence multiplied by their corresponding volume fraction. The general form of the birefringence is then written as

Figure 8. Field-induced birefringence normalized by the cellulose volume fraction, plotted vs magnetic induction B = μ0H (in Tesla) . The experimental data (full circles) are compared to fits obtained from calculated magnetic birefringence curves (lines) in the approximation of a monodisperse aqueous suspension of average tunicin nanocrystals (from the size distribution Distrib2), for the corresponding models. Note that the curves of model 3 are intermediate between the two extreme cases of model 1 and model 2.

Δn fit(H ) Φ ∞

=

∫ ∫ ∫0 LaLbLcPabc(La , Lb , Lc)δn0 ·S2(γ ) dLa dLb dLc ∞

∫ ∫ ∫0 LaLbLcPabc(La , Lb , Lc) dLa dLb dLc (25)

= 74 × 10−6) but one can see that the corresponding fitting curves does not match well the experimental (black) curve. In the case of Distrib2 the best fit is obtained with model 1, with −6 αmono = 0.93 and Δχmono (Qfit = 73 × 10−6). fit a(bc) = −2.65 × 10 In these two monodisperse approximations, the specific birefringence of cellulose nanocrystals is calculated, using eqs 21−23 and S2 = 1, to be δn0 = +0.121. This value compares to δnintr 0 = +0.0744 for the intrinsic birefringence of bulk native cellulose55 and illustrates the importance of taking into account

Here Pabc(La, Lb, Lc) is a trivariate size distribution accounting for combined normal and log-normal distributions of La, Lb and Lc, as well as the possible correlation between each other. The term δn0 is under the integral as its dependence on the crystal dimensions is implicit and has been calculated for each (La, Lb, Lc) trinome using eqs 21-23. In practice, we performed a numerical integration over (La, Lb, Lc) using the following discrete sum:

Table 2. List of Δχ [10−6 SI] for the Investigated Models within the Two Distributions, Fitting the Mono- and Polydisperse Distributionsa

a

Distrib

model

r

ρac

fit

αfit

Δχzy

Δχyx

Δχa(bc)

δn0

Qfit × 10−6

Veff (nm3) × 105

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2

1 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3

0 ∞ 1 0.727 0 ∞ 1 0.727 0 ∞ 1 0.727 0 ∞ 1 0.727 0 ∞ 1 0.727 0 ∞ 1 0.727

− − − − 0 0 0 0 ρab ρab ρab ρab − − − − 0 0 0 0 ρbc ρbc ρbc ρbc

mono mono mono mono poly poly poly poly poly poly poly poly mono mono mono mono poly poly poly poly poly poly poly poly

0.93 0.89 0.91 0.91 0.997 0.95 0.97 0.98 1.01 0.968 0.99 0.99 0.93 0.916 0.916 0.92 1.02 0.97 1.00 1.00 1.03 0.99 1.01 1.01

−5.85 0 −3.63 −4.105 −3.10 0 −1.97 −2.22 −2.44 0 −1.55 −1.73 −2.65 0 −1.65 −1.86 −1.15 0 −0.73 −0.82 −0.95 0 −0.60 −0.68

0 −8.11 −3.63 −2.98 0 −4.63 −1.97 −1.60 0 −3.64 −1.55 −1.26 0 −3.68 −1.65 −1.35 0 −1.72 −0.73 −0.59 0 −1.43 −0.60 −0.49

−5.85 −4.05 −5.45 −5.59 −3.10 −2.32 −2.95 −3.01 −2.44 −1.82 −2.32 −2.37 −2.65 −1.84 −2.47 −2.54 −1.15 −0.86 −1.10 −1.12 −0.95 −0.71 −0.91 −0.93

0.121(1) 0.121(1) 0.121(1) 0.121(1) 0.121(0) 0.121(0) 0.121(0) 0.121(0) 0.121(2) 0.121(2) 0.121(2) 0.121(2) 0.120(3) 0.120(3) 0.120(3) 0.120(3) 0.120(0) 0.120(0) 0.120(0) 0.120(0) 0.120(2) 0.120(2) 0.120(2) 0.120(2)

74 209 110 99 5.14 23.3 8.78 7.5 3.3 15.9 5.5 4.6 73.6 208.9 110 98.6 3.03 14.9 5.02 4.27 2.44 11.04 3.6 3.11

1.17 1.17 1.17 1.17 1.28 1.28 1.28 1.28 1.43 1.43 1.43 1.43 2.58 2.58 2.58 2.58 3.11 3.11 3.11 3.11 3.38 3.38 3.38 3.38

The best fit has been highlighted in bold. J


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Macromolecules Δnfit(H ) 1 = αfit Φ Vave

The total birefringence can finally be normalized by δnsat to get the average saturation value in the sample, which is around 88% at H = 140,000 kA/m (i.e., B ≈ 17.5 T). 3. Determination of the Diamagnetic Anisotropy of the Cellulose Nanocrystals from that of its Individual Bonds. We have evaluated the effective diamagnetic susceptibilities along the three principal crystalline axes (x, y, z) of the cellulose crystal, where x and y are, respectively, perpendicular and parallel to the mean plane of the AGU ring and z is along the cellulose chain axis (Figure 10), using Pascal’s

xamax xbmax xcmax

∑ ∑ ∑ LaLbLc Pabc(La , Lb , Lc) xamin xbmin xcmin

...(− δn0 /2)·Σ 2

(26)

xamax xbmax xcmax

Vave =

∑ ∑ ∑ LaLbLc Pabc(La , Lb , Lc) xamin xbmin xcmin

(27)

Here we introduced linearly spaced xi defined for the LNN type case as xa = ln(La), xb = Lb, and xc = Lc or for the LLN type case as xa = ln(La), xb = ln(Lb), and xc = Lc and we have chosen to map both distributions with 50 × 50 × 50 = 125,000 points (cf. Supporting Information, sections 1 and 2). The experimental birefringence curve is fitted for each model in Figure 9. The corresponding fitting parameters are also listed

Figure 10. Orientation of the x, y, z axes with respect to a cellobiosyl motif of the cellulose crystal.

principle of additivity of the individual covalent bond anisotropies. The three principal anisotropies of the crystal are given by χzz ′ −χyy ′ , χyy ′ − χxx ′ and χxx ′ − χzz ′. From the details of the calculations that are presented in the Supporting Information, Section 4, we obtain the following diamagnetic anisotropic susceptibility for cellulose Iβ.

Figure 9. As in Figure 8, but taking into account the experimental polydispersity.

Δχzy′ = χzz′ − χyy′ = −0.49(9) × 10−6 SI ′ = χyy′ − χxx ′ = −0.36(3) × 10−6 SI Δχyx

in Table 2 where the correlation ρac postulated in this work was either ρac = ρab (LNN type) or ρac = ρbc (LNN) type. In the case of distribution 1, the best fit is obtained for model 1 with αpoly fit = −6 (Qfit = 3.3 ×10−6). For 1.01 and Δχpoly a(bc) = −2.44 × 10 distribution 2, the best fit is again obtained with model 1 with −6 poly (Qfit = 2.44 × 10−6). αpoly fit = 1.03 and Δχa(bc) = −0.95 × 10 The fact that we have approximated the nanocrystals by an elliptical cross section may induce a small error in overestimating the section by about 10%, thus underestimating the Δχij by 10% in the case of distribution 1. In order to determine the saturation value of the birefringence of our sample, we have also recalculated eq 27 by setting the S2 parameter to −1/2 (when H → ∞). As reported in Table 2 this calculation led to specific and saturation birefringence:

′ = χxx ′ − χzz′ = + 0.86(3) × 10−6 SI Δχxz

(28)

These values indeed show that, the cellulose crystals will tend to orient biaxially, with their long axis (z) perpendicular to H and the normal to the mean sugar planes (x) parallel to H. rPascal =

′ Δχyx Δχzy′

≈ 0.727 (29)

The ratio r = 0.727 is therefore expected according to this analysis. From the above values, one can calculate an in-plane (b,c) averaged uniaxial anisotropy: Δχa′(bc) = Δχz′(xy)

δn0 = +0.120(2) and δnsat = −0.060(1)

′ + Δχzy′ )/2 = (Δχzx

An estimation of the birefringence of aqueous dispersion of TEMPO treated wood cellulose microfibril has been earlier proposed by Lasseuguette et al.62 to δn0 = 0.123. This value was estimated by summing the intrinsic birefringence of cellulose crystals (δnintr 0 = +0.08) from ref 63 and a shape contribution (δnshape = +0.043) using Wiener formula,64 assuming an average 0 optical index of 1.56 for cellulose and 1.33 for water. As we have followed a more rigorous approach in δn0 derivation, we believe the value of δn0 = +0.120 is more accurate for our tunicin nanocrystals.

= −0.68(5) × 10−6 SI

(30)

The absolute value of this anisotropy is slightly lower (by a factor ∼1.4 smaller) than the one deduced from the above experimental value Δχ = −0.95 × 10−6, taking into account distribution 2 (assuming ρac = ρbc) and model 1, which stands out as being the best. It is important to note that a slight error on the determination ot the dimensions will directly affect the absolute value of the susceptibility measured by the birefringence experiment and it could technically explain any K


Article

Macromolecules mismatch. To illustrate this, one can note that assuming ρac = 0 with distribution 2 provides a value Δχ = −1.15 × 10−6 about ∼20% larger. However, modifying the size distribution cannot affect the ratio r = 0.727 derived from Pascal’s additivity law, which is therefore a rather robust quantity.

lattice are nearly equivalent in both cellulose Iα and Iβ.66,67 Thus, the Δχ value calculated from the Pascal’s law of additivity for cellulose Iα should be nearly the same as the one calculated for cellulose Iβ in the above section 3. From this study, we could also deduce the specific birefringence of tunicin nanocrystals in their water environment. A value of δn0 = +0.120(2) was obtained. This value is well accounted for by considering both the intrinsic = +0.074455 and a significant birefringence of cellulose δnintr 0 contribution from the slender shape of the nanocrystals as calculated according to eqs 21−23. It turned out rather robust with respect to the three different models used (cf. Table 2, δn0 and αfit). In calculating the diamagnetic anisotropy applying the Pascal’s additivity law on the cellulose crystal structure, we obtained the values Δχzy = −0.49(9) 10−6 SI, Δχyx = −0.36(3) 10−6 SI and Δχxz = +0.86(2) 10−6 SI. These values account for the fact that that the cellulose nanocrystals orient not only perpendicular to the magnetic field, but should also rotate about their long axis, tending to align the mean plane of the cellulose AGU rings perpendicular to the field direction. Such a specific biorientation can be qualitatively explained by the fact that each of the four C−C segments and the two C−O segments that constitute the glucosyl ring in its 4C1 chair conformation, all have negative diamagnetic anisotropy values.68,69 In addition, the five moieties that are attached to the glucosyl rings, namely C5−C6, C2−O2, C3−O3, C1−O1, and C4−O1 are all in an equatorial conformation and thus reinforce further the torque leading to the perpendicular alignment of the AGU ring mean plane with respect to the field. The five C−H moieties, which are in axial position, would tend to counterbalance this perpendicular alignment, but the diamagnetic anisotropy contribution of these segments68 is far lower than that of the other segments and therefore the effect of five axial C−H appears to be negligible. The remaining five bonds of low diamagnetic anisotropy, namely the O6−H, O3−H, O2−H, C6−Ha, and C6−Hb are not aligned with respect to the axial or equatorial positions of the glucosyl ring and they are expected to contribute only a little to the overall alignment. Same reasoning applies for the C6−O6 bond, which is inclined with respect to the three axes of the cellulose crystals. Given these features and the Δχ values deduced from the fitted birefringence curves, model 3 would be expected to lead to better fit than model 1, but in fact, it is not the case, as model 1 leads explicitly to a better fit of the birefringence curves (see the Qfit values in Table 2 as well as the fit in Figure 9). We believe that the lack of biaxial orientation within the field can be related to a specific characteristic of tunicin nanocrystals that are renown to continuously twist with a halfhelical pitch of 1.2−1.6 μm when in suspension.54 The twisting occurring in these long crystals does not interfere with the orientation of the nanocrystals with their axes perpendicular to the field, but it prevents them from orienting along the secondary orientation expected from the biaxial characteristics. While such twist is expected to entirely average out the physical properties of χxx and χyy of rods as long as a quarter-pitch distance, i.e., 0.6−0.8 μm, biaxial orientation should be relevant for shorter cellulose nanocrystals, such as those extracted from cotton and wood. The determination of diamagnetic anisotropy values, which is the salient result of this report, is important, since the use of magnetic orientation is of increasing interest for the processing of magnetically oriented cellulose-based products.37−43 In

■

DISCUSSION In this study, we have experimentally verified the negative sign of the diamagnetic anisotropy of native cellulose nanocrystals, when defined as usually by comparing their long axis to the two others. As a consequence, rod-like cellulose nanocrystals tend to orient themselves with their long axes perpendicular to a strong magnetic field. Our results confirm our earlier work34,35 but also the study of Loeb and Welo32 and that of Nilakantan33 who both showed the same effect, but using different cellulose samples. On the other hand our results seem to contradict some of the results of Cotton-Feytis and Fauré-Frémie31 who suggested a positive value for a number of fibers ranging from cotton to nettle and retted flax. Interestingly, these authors obtained, as we did, a negative value in the case of fibers from retted hemp. Our results were achieved on suspensions of noninteracting and isolated rod-like single nanocrystals are the basic structural elements of native tunicin. We believe that our results on these cellulose nanocrystals are more reliable than those on entire fibers where the orientation of the axis of the individual crystals is frequently different from that of the cellulose fibers. In earlier studies, we revealed the perpendicular orientation of the nanocrystals with respect to the magnetic field by electron diffraction34 and optical microscopy.35 The present report goes one step further in providing both an experimental and calculated value for the diamagnetic anisotropy Δχ of native cellulose. Obtaining the value of Δχ from the variation of the birefringence of the nanocrystal suspension as a function of the magnetic field strength was found to be highly dependent on the measurement of the dimensions of the cellulose nanocrystals. In fact, not only the distribution of their lengths, widths and thicknesses were important, but also the correlation between these dimensions had to be accounted for. When considering only the crystalline part of the nanocrystals, as revealed by diffraction contrast transmission electron microscopy, a value of Δχz(xy) = −2.44 × 10−6 (in SI units) was obtained. On the other hand, if the overall volume of the nanocrystals, including their surface chains, was considered, as revealed from negatively stained electron micrographs,54 a more relevant value of Δχz(xy) = −0.95 × 10−6 resulted from the birefringence curve fitting. In absolute value this Δχ is somewhat higher than a theoretical value of Δχz(xy) ′ = −0.68(5) × 10−6, that we estimated from the cellulose crystalline coordinates, using Pascal’s additivity law. The difference between these two values may result from the uncertainties both in the measurements of the nanocrystal widths and thicknesses but also in the standard literature values taken for the diamagnetic anisotropy of the individual chemical bonds constituting the cellulose molecules. It is interesting to note that the present work was achieved with suspensions of tunicin nanocrystals that consist essentially of the Iβ allomorph of native cellulose.65 Regarding the other allomorph, namely cellulose Iα, which is found in various proportions in most other cellulosic organisms, it presents a different unit cell and the cellulose chains are packed with different symmetry elements. Nevertheless the conformation of the cellulose chains and their organization within the crystalline L


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Macromolecules particular, the knowledge of the Δχ for cellulose is critical to devise the properties of these products, by choosing appropriate cellulose sources and selecting magnetic field strengths. In literature, the Δχ value for cellulose fibers has been estimated in the range from −10−7 SI32,49 to −10−8 SI.50 The Δχ that we are presenting here are slightly different since they were obtained with isolated cellulose nanocrystals that differ from the complex structures found in fibers. The main advantage of the present work is that it can be applied directly to the standard crystal of cellulose Iβ. In the course of this study, we also analyzed less crystalline samples of cellulose. One example was that of wood cellulose nanocrystals that have lengths nearly 10 times smaller than those of tunicin, but comparable lateral dimensions. Their dilute suspension, prepared following the above protocol also showed perpendicular orientation into the strong fields, but there was no saturation of the birefringence that could be observed up to the highest field (17.5 T). Another investigated system was that of dilute non flocculated suspensions of crab chitin nanocrystals prepared following the method described by Li et al.70 These nanocrystals, which had dimensions comparable to those of wood cellulose were also found to orient perpendicular to the magnetic field, but due to the lack of reliable values for the intrinsic birefringence of crystalline chitin, we could not carry out an equivalent analysis to obtain the value of their negative diamagnetic susceptibility. Nevertheless, using the coordinates of crystalline α-chitin71 and the standard values for the anisotropy of individual bonds, including N−H and CO, preliminary calculations using the Pascal’s law of additivity led to a negative Δχ for α-chitin with a value similar to that of cellulose and again showed the propensity of α-chitin to biaxially orient when subjected to magnetic field. Finally, we also studied aqueous suspensions of amylose/α-naphthol single crystals, prepared following the method of Cardoso et al.72 Unlike cellulose and chitin, which had a negative Δχ with respect to their long dimension, these crystals had a positive Δχ, a feature that might be due either to the 8-fold helical organization of amylose or that of the αnaphthol intrahelical guest molecules.

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using Pascal’s additivity law applied to a perfect crystal of cellulose Iβ, the cellulose allomorph exclusively present in tunicin. The estimation from the Pascal’s law also yielded a positive value for Δχxz indicating that in addition to its alignment perpendicular to the field, perfect cellulose Iβ crystals should also rotate so that the mean plane of the cellulose AGU rings, should become perpendicular to the field direction. This double orientation could not be deduced from our Cotton−Mouton data, likely due to the twisting of the tunicin nanocrystals about their axis. It remains to be seen whether nontwisted cellulose crystals, when in suspension, would orient biaxially under strong magnetic field.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02201. Details about the correlated trivariate size distributions (sections 1 and 2), the fast computation of the function Σmix 2 (section 3), and the derivation of Pascal’s principle of additivity of the covalent bond anisotropies to determine cellulose diamagnetic anisotropic susceptibility (section 4) (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*(B.F.-P.) E-mail: [email protected]. *(H.C.) E-mail: [email protected]. Present Address ∥

Melville Laboratory for Polymer Synthesis, Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, U.K. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support for this work in the framework of the CNRS-JSPS joint research program is acknowledged. The authors also thank the Grenoble High Magnetic Field Laboratory, Max-Planck- Institut für Festkörperforschung and CNRS for the use of one of their magnets. The authors thank L. Heux for valuable suggestions during the writing of this work. They are also grateful to Y. Nishiyama for providing the raw data from ref 54.

CONCLUSIONS

In this work, we have measured the birefringence variation of dilute aqueous suspensions of nonflocculating and noninteracting slender tunicin nanocrystals subjected to increasing magnetic fields (Cotton−Mouton measurements). When reaching a maximum field of 17.5 T, the birefringence was maximized, indicating that nearly all the nanocrystals were oriented with their long axis perpendicular to the field. The analysis of the birefringence data and the knowledge of the polydispersity of the lengths, widths and thicknesses of the nanocrystals, together with the correlations existing between these three parameters allowed: 1 To obtain the specific birefringence of tunicin nanocrystals suspended in water, which was found to be δn0 = +0.120(2). 2 To calculate the value of the diamagnetic susceptibility Δχz(xy) = χzz−(χxx + χyy)/2 of tunicin cellulose, with a value of −2.44 × 10−6 if only the crystalline core of the nanocrystals was considered or of −0.95 × 10−6 if their whole width was accounted for. These values can be compared with Δχz(xy) ′ = −0.68(5) × 10−6 estimated by

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REFERENCES

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