Generalized Indirect Fourier Transformation as a Valuable Tool for the

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Generalized Indirect Fourier Transformation as a Valuable Tool for the Structural Characterization of Aqueous Nanocrystalline Cellulose Suspensions by Small Angle X‑ray Scattering Heike M. A. Ehmann,†,§,‡ Stefan Spirk,*,†,‡,∥ Aleš Doliška,†,‡ Tamilselvan Mohan,†,‡ Walter Gössler,§ Volker Ribitsch,§,‡ Majda Sfiligoj-Smole,†,‡ and Karin Stana-Kleinschek†,‡ †

University Maribor, Faculty of Mechanical Engineering, Institute for Engineering Materials and Design, Laboratory for the Characterization and Processing of Polymers, Smetanova Ulica 17, 2000 Maribor, Slovenia § Karl-Franzens-University Graz, Institute of Chemistry, Heinrichstraße 28, 8010 Graz, Austria ∥ University of Technology Graz, Institute for Chemistry and Technology of Materials, Stremayrgasse 9, 8010 Graz, Austria ABSTRACT: Small angle X-ray scattering (SAXS) is employed to characterize the inner structure and shape of aqueous nanocrystalline cellulose suspensions using the generalized indirect Fourier transformation (GIFT). The use of the GIFT approach provides a single fitting procedure for the determination of intra- and interparticle interactions due to a simultaneous treatment of the form factor P(q) and the structure factor S(q). Moreover, GIFT allows for the determination of particle charges and polydispersity indices. As test material, aqueous nanocrystalline cellulose suspensions (aNCS) prepared by the H2SO4 route have been investigated and characterized (SAXS, dynamic light scattering, zeta potential).

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capacitors,6 it is crucial to understand the structure−property relationships of aNCS-based materials. While techniques such as atomic force microscopy (AFM) can only give insight into the outer shape and the morphology of the NCC deposited on surfaces, scattering techniques such as static light scattering, small-angle X-ray scattering (SAXS), or small angle neutron scattering (SANS) are able to provide a deeper insight into the inner structure of aNCS. In case of SAXS, the inverse Fourier transformation (IFT) is widely used for the evaluation of small angle scattering data. However, the main disadvantage of IFT is its limitation to diluted systems, where interparticle interactions are considered to be negligible due to an essentially larger average distance between the particles compared to their dimensions. For more concentrated solutions, this assumption is not fully valid, and therefore, interparticle scattering must be taken into consideration for data evaluation. Usually, intraparticle scattering is expressed by the so-called form factor P(q), while interparticle scattering is summarized in the structure factor S(q) where q is the length of the scattering vector. The product of these two parameters is represented in an approximation by the total scattering intensity I(q). The influence of S(q) can be seen at low q values, where the scattering curves exhibit a remarkable deviation from an ideal

INTRODUCTION In the past decades, there has been growing interest in the preparation and applications of nanocrystalline cellulose (NCC) due to their unique physical and chemical properties.1,2 While their high mechanical strength has been exploited for the preparation of mechanically reinforced materials, their nonlinear optical behavior has been the subject of many investigations toward the design of smart materials.3,4 The strategies for the synthesis of NCC are based on the differences in solubility and chemical reactivity of the amorphous and crystalline entities of bulk cellulose toward acidic media. In particular, the amorphous parts of bulk cellulose are preferably dissolved and hydrolyzed upon exposure to acidic solutions, while the crystalline domains readily reassemble to form nanocrystals, nanorods, or nanowhiskers, for instance.5 In the course of the preparation of cellulose nanocrystals, sulfuric acid is the most commonly used acid.1 This agent not only removes the amorphous parts of bulk cellulose by dissolution and hydrolysis, but also reacts with the remaining crystallites to create sulfate groups on their surface by ester bond formation. As a result, the nanocrystals are highly negatively charged, which is advantageous regarding their stability in aqueous suspensions according to the DLVO theory. In parallel to investigations on the preparation of NCC, the elucidation of their inner structure has become an important topic during the past years. Especially, for the design of smart materials such as electronics,4 smart papers, and super© 2013 American Chemical Society

Received: August 1, 2012 Revised: February 1, 2013 Published: February 21, 2013 3740

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The data evaluation and correction as well as all simple mathematics (e.g., subtractions, multiplications, and data binning) were performed using the software program PDH (Primary Data Handling) from the PCG software package version 3.01.10.11 Further, the different extrapolations (Guinier, Porod, Invariant) were also performed with the software PDH. The indirect Fourier transformation as well as the generalized indirect Fourier transformation were performed using the software program GIFT (Generalized Indirect Fourier Transformation) from the PCG software package version 3.01.10.11 The Lagrange multiplier was determined using the interval from λ = 10 to 0, whereas the search was performed in 0.25 steps which corresponds to 41 Lagrange multipliers. To obtain 201 equidistant points in the r-scale, 20 cubic B-splines were used as basis functions. As minimization algorithm, the Boltzmann simplex simulated annealing algorithm with a start annealing temperature of 20 °C was used. In the case of GIFT4, a core shell approach was employed. For this purpose, a core of with a radius of 40 nm was assumed with a shell of 1 nm (contrast 1:10). For the rod-like approach, a core of 19 nm and a shell of 1 nm were used (contrast 1:10). The salt concentrations for the models GIFT5 to 7 were set to zero and the volume fraction was kept constant in the calculations for the determination of the other two linearly dependent parameters (r, Zeff). The only exception represents the cross section approach where for GIFT5 a reasonable fit could not be obtained. Therefore, the radius r was kept constant for this model in order to determine the other two parameters (volume fraction, effective charge). The model parameters are summarized in Table 1.

particle scattering curve behavior. This results in a decrease in total scattering intensity at low q values due to repulsive interactions between the particles which is followed by a more or less pronounced increase. These effects at low q values have a significant impact on the evaluation of experimental data. Although there are several approaches known in the literature to overcome this problem (e.g., concentration dependent measurements and extrapolation to zero intensity, elimination of the inner part of the scattering curve), in our opinion, the most convenient method is to use the so-called generalized indirect Fourier transformation (GIFT).7−10 In principle, this method offers several advantages: the data evaluation can be done for concentrated solutions, it requires only a minimum amount of a priori information, it is capable of determining P(q) and S(q) simultaneously, and the influence of particles with charged groups can be evaluated. In this work, we will focus on the applicability of the GIFT approach to determine structural parameters of aNCS derivatives and we will outline several advantages over “simple” IFT methods.

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EXPERIMENTAL SECTION

Materials. As cellulose source, microcrystalline cellulose (Sigmacell Cellulose Type 20 from Sigma, cotton source) was used. Sulfuric acid (concentrated) was supplied from Carl Roth with a purity of 96% Rotapuran. Dialysis was performed using regenerated cellulose membranes (Carl Roth) with a molecular weight cutoff of 8000−10 000 Da. The AFM measurements were performed using silicon wafer from SilChem. These Si-wafers have been precoated using polyethyleneimine (20 mM in 0.5 M KCl, branched, high molecular weight, SigmaAldrich). Preparation of Aqueous Nanocrystalline Cellulose Suspension (aNCS). The aqueous nanocrystalline cellulose suspensions (aNCS) are prepared using sulfuric acid (4.2 M) to hydrolyze microcrystalline cellulose (MCC). In a typical procedure, MCC (1.0 g) is treated with 10 mL of acid under stirring and subsequently heated to 45 °C. The reaction is stopped by adding 90 mL of distilled water, cooled to 4 °C, and subjected to centrifugation (10 min at 11 000 rpm). The precipitate is redispersed in doubly distilled water and centrifuged. This procedure is repeated until the pH of the suspensions reaches a value higher than 4. Purification is done by dialysis against doubly distilled water for one week, whereby the water is exchanged every 24 h. After dialysis, the suspensions are diluted to a concentration of 1.0 wt % and redispersed by sonication using a VibraCell VCX 750 (sonicator with a high-intensity Ultrasonic Processor) ultrasonication dip for 5 min using 0.5 s pulses with 1.5 s intervals at a fixed amplitude of 30%. Small Angle X-ray Scattering. The SAXS experiments were performed using a S3-MICROpix solution of Hecus with a 50 W microsource Genix 2009 from Xenocs. The tube consists of a copper anode with an emission wavelength of 1.5418 Å for the Kα line. The sample to detector distance was 291 mm with an angle of 4.2°. The optics are 3D for point focus with a beam size of 50 × 200 μm2 and a flux up to 4 × 108 photons s−1 mm−2. The point focus at the detector has a monochromatic SAXS resolution of q(min) ≥ 4 × 10−3 Å−1. The scattering vector (q) range is between 0.003 and 1.9 Å−1. As a detection system, a 2D Pilatus 100k Dectrics Detector 34 × 84 mm2, with a pixel size of 172 × 172 μm2, was used. A Nickel filter was used as semitransparent primary beam stop. The aNCS concentrations were adjusted to 1.0 wt %. The samples were sealed in a quartz glass mark tubes of Hilgenberg with a linear adsorption coefficient of 75.8 cm−1 for CuK L-radiation. The outer diameter of the mark tubes was 1.0 mm with a wall thickness of 0.01 mm. The samples were irradiated with 50 kV and 1 mA emission current for 1800 s. The background measurement was done also for 1800 s but with closed shutter.

Table 1. Model Parameter Intervals for the GIFT Calculationsa

0 1 2 3 4 5 6 7

IFT GIFT1 GIFT2 GIFT3 GIFT4 GIFT5 GIFT6 GIFT7

volume fraction

radius [nm]

polydispersity

effective charge

0.000−0.640 0.000−0.640 0.000−0.640 0.000−0.640 0.000−0.640 0.000−0.640 0.000−0.640

0.0−50.0 0.0−50.0 0.0−50.0 0.0−50.0 0.0−50.0 0.0−50.0 0.0−50.0

0.000−1.000 0.000−1.000 0.000−1.000 0.000−1.000 0.000−1.000 0.000−1.000

0−1000 0−1000 0−1000

a

GIFT 4: 40 nm core with 1 nm shell, contrast core:shell is set 1:10. GIFT4 cs: 19 nm core with 1 nm shell, contrast core:shell is set 1:10.

Atomic Force Microscopy. Sample surfaces were characterized by atomic force microscopy (AFM) in acousting AC mode (tapping mode) with an Agilent AFM 5500 system (Agilent Technologies, USA). The images were scanned using silicon cantilevers (ATEC-NC, Nanosensors, Germany) with a resonance frequency of 210−490 kHz and a force constant of 12−110 N m−1. All measurements were performed at room temperature under ambient air. The data evaluation was performed using the freeware Gwyddion 2.26. Dynamic Light Scattering (DLS) and ζ-Potential. The DLS experiments were performed using ZetaPALS Zeta Potential Analyzer Utilizing phase Analysis Light Scattering from Brookhaven Instruments which is equipped with a class I laser (35 mW, red diode laser, λ = 660 nm). The scattering angle is set to 90°. Inductively Coupled Plasma Mass Spectrometry. Aliquots (∼50 mg) of the dried nanocrystalline cellulose sample were weighed to 0.1 mg into 12 mL quartz digestion vessels and 5 mL of HNO3 were added. The samples were heated in the UltraClave IV (EMLS, Leutkirch, Germany) using the following program: Step 1: 5 min → 80 °C; step 2: 15 min → 150 °C; step 3: 15 min → 250 °C; step 4: 30 min at 250 °C. For the S-determination, an Agilent 7500ce at m/z 34 system was used. The instrument was tuned to give 9.0 × 105 cps at m/z 7 for a 1 μg Li/L, 10 × 105 cps at m/z 89 for a 1 μg Y/L solution, and 7.5 × 105 cps at m/z 205 for a 1 μg Tl/L solution. The oxide ratio 156 CeO/140 Ce was less than 0.013. As internal standard Be at m/z 9 3741

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was used. Data acquisition and evaluation were performed with ICPMS Masshunter B.01.01 (Build 123.10 Patch 3) software.

In the literature, it is quite common to define the resolution (the maximum particle dimension D) from the lowest scattering angle qmin using the Bragg relation in eq 7.

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THEORETICAL BACKGROUND SAXS experiments are realized by monitoring the intensity of the scattered photons at small angles θ. The modulus of the wavelength-dependent scattering vector q can be expressed according to eq 1. |q| =

⎛ 4π ⎞ ⎜ ⎟sin θ ⎝ λ ⎠

qminDBragg = 2π

The scattering intensity I(q) is related to the averaged ensemble form of the particles. These intraparticle interferences can be separated by the so-called form factor P(q). Further, the interparticle ordering is related to the scattering curve and can be expressed by the structure factor S(q). The ordering originates from the interaction potential of the particles and is interrelated with the shape of those. In diluted and uncharged systems, S(q) can be neglected. The scattering intensity for N particles can be expressed by eq 2 for monodisperse particles, with a homogeneous spherical shape.

0

∫∞ p(r)

(2)

sin(qr ) dr qr

Rg2 =

Lπ Ic(q) q

∫0

2

(3)

Rg2 =

∫A

Δρc (r′)Δρc (r′ + r ) dr c

2

(10)

∫ p(r )r 2 d r 2 ∫ p(r ) d r

(11)

For rod-like particles, the radius of gyration of the cross section Rc can be determined on the one hand directly via the so-called cross section Guinier plot (log(I(q)q) vs q2) according to eq 12 or on the other hand from the PDDF of the cross section pc(r) according to eq 13. 2

2

Ic(q) = Ic(0) eq R c /2

∞

pc (r ) = γc(r )r = 2πr

(9)

Rg can be directly determined via the so-called Guinier-plot (ln(I(q)) vs q2, the slope is proportional to Rg2). The following expression, eq 11, indicates the direct relation between Rg and the pair distance distribution function (PDDF), once it is known.

(4)

pc (r )J0 (qr ) dr

∫ Δρ(r1)ri2 dVi ∫ Δρ(ri) dVi

I(q) = I(0)eq R g /3

Here, L is the rod length and Ic(q) is the cross section scattering function and is related to the cross section distance distribution function pc(r) in eqs 5 and 6. Ic(q) = 2π

(7)

The second moment of the function in one space is related to the curvature (second derivative) of its Fourier transform at the origin. This is according to the momentum theorem of Fourier transformation and the basis of the so-called Guinier approximation (eq 10)

The scattering function of a rod-like particle with the length L and the cross-section Ac (with the maximum dimension d) can be described by the following eq 4. I(q) =

2π qmin

For diluted systems, a direct relation does exist via Fourier transformation between the particles themselves and the scattering curve. If the concentration increases and hence the interparticle interaction, the structure factor has to be taken into account.15,20−22 Radius of GyrationRg. The radius of gyration is one of the most important parameters in SAS experiments (eq 9).

For polydisperse systems, this separation is no longer true and the structure factor has to be replaced by the so-called effective structure factor Seff.8,12−14 The distance distributions of the overall mean value density fluctuations are described by the total distribution function i(r). Hence, these density fluctuations are related to the particle density distributions and described by the correlation function g(r). The total correlation function h(r) describes the deviation of particle distribution from the statistical uncorrelated point of view.15 IFTIndirect Fourier Transformation. The evaluation of the data of small angle scattering (SAS) experiments with IFT leads to the determination of the pair-distance-distribution function p(r).16 The p(r) is related to the spatially averaged intensity I(q) as given in eq 3. It contains the information on the size, shape, and inner structure.17 I(q) = 4π

DBragg =

This is true for periodical structures, but not for isolated particles with a finite extension.18 DBragg is only a measure for the smallest scattering angle which is not identical to the maximum dimension of the particles. According to the sampling theorem of Fourier transformation, the scattering curve is observed at increments Δq ≪ qmin; the scattering data contains the full information for all particles with the maximum dimension D (eq 8). π D≤ qmin (8)

(1)

I(q) ∝ NP(q)S(q)

or

Rc2 =

(5)

∫ pc (r )r 2 dr 2 ∫ pc (r ) dr

(12)

(13)

It should be mentioned that the cross section intensities of rod-like particles Ic(q) drop to zero in the Guinier range. This is caused by the finite length of those particles. Hence, the accurate determination of the radius of gyration of the cross section Rc via the Guinier approximation is not possible.18 GIFTGeneralized Indirect Fourier Transformation. The GIFT procedure8 can be regarded as the generalization of

(6)

Here, γc(r) is the spatial autocorrelation function, Δρc is the fluctuation of the electron density. The size range, which can be analyzed by small angle scattering experiments, ranges from a few nanometers up to hundred nanometers depending on the quality of the setup.18,19 3742

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the IFT.16,23,24 Although a brief overview on the approach is given in the following, more interested readers are referred to the detailed description of the method.8−10 The GIFT approach is able to determine both the structure and the form factor simultaneously. While the form factor P(q) can be determined model free, the structure factor S(q) has to be calculated according to an appropriate model. The transformation of the scattering curve I(q) into real space without separating S(q) leads to a hardly interpretable Fourier transformation resulting in strong oscillations at high values of r.9 The structure factor can be determined by a nonlinear leastsquares procedure including linearization of the fit function. This inefficient nonlinear least-squares routine is replaced by the Boltzmann simplex simulated annealing (BSSA). The BSSA algorithm is a modified version of the downhill simplex algorithm (DS). It can be used to optimize a local minimum of an arbitrary function. In GIFT calculations, there is an expansion by a term for S(q) as shown in eq 14

Here, P(q) is the averaged form factor, Seff(q) is the effective structure factor, fα the form amplitude of species α at q = 0, Bα(q) is the normalized form amplitude of species α (Bα(0) = 1) and Sαβ(q) is defined as the partial structure factor. The real polydispersity has a huge influence on the structure factor at zero scattering vector S(0), and the form factor can be varied for homogeneous or inhomogeneous particle systems. In the following, the models used for the calculation of the structure factor are described. Structure Factor Models. GIFT1: Hard Spheres, Monodisperse (PY). The structure factor is calculated according to the Percus−Yevick (PY) closure relation (eq 19), whereby the thermodynamic β = 1/kT with k being the Boltzmann constant and υ(r) the interaction potential.25 g (r ) = e−βυ(r)[g (r ) − c(r )]

The interaction effects in this model are based only on the excluded volume which is present in a dispersion of hard spheres. This model can be used for any repulsive interaction by means of an effective hard sphere approximation. PY closure calculations are very fast but inaccurate for charged particles. GIFT2: Hard Spheres, Polydisperse (PY). The structure factor is calculated according to the Percus−Yevick (PY) closure relation in eq 19, and again, only the excluded volume which is present in a dispersion of hard spheres is taken into account for the calculations.25 The polydispersity is considered by simply averaging the partial structure factors of the single components.8 This model is suitable for spherical and slightly elongated uncharged particles. Therefore, for charged particles this model will lead to wrong parameters, even if the p(r) function is acceptable. GIFT3: Homogeneous Hard Spheres, Polydisperse (Effective PY, Schulz Distribution). The structure factor for polydisperse uncharged spheres is calculated according to the Percus−Yevick (PY) closure relation, eq 19.25 The calculated polydispersity is theoretically correct according to Weyerich et al., but in general, the formalism is limited to real hard spheres.9 GIFT4: Inhomogeneous Hard Spheres, Polydisperse (Effective PY). The structure factor is calculated according to polydisperse uncharged spheres,9 whereas the form factor has to be known in advance. GIFT5: Charged Spheres, Monodisperse (Yukawa Potential, RMSA Closure). The structure factor for charged hard spheres via the rescaled mean-sphere approximation (RMSA)26 can be calculated using the Yukawa potential in eq 20.

n

I(q) = P(q)S(q , dN ) = [∑ cvψv(q)]S(q , dn) ν= 1

(14)

Hereby, cv are the expansion coefficients, ψ are the transformed splines, n is the total number of splines, and dN is a vector which contains the structure factor parameters. The product of the S(q) with expansion coefficients renders the approximation nonlinear. Hence, it excludes the use of any simple linear least-squares algorithms. A detailed description to solve this highly nonlinear problem can be found elsewhere.10 As shown in eq 2, the scattering intensity I(q) for N-particles can be split into the form factor P(q) and the structure factor S(q). Then, the form factor P(q) is related to the pair distance distribution function p(r) as described in eq 15. P(q) = 4π

∫0

∞

p(r )

sin(qr ) dr qr

(15)

In eq 16, the relation between the structure factor [S(q) − 1] and the total correlation function [g(r) − 1]r2 is shown. S(q) − 1 = 4πn

∫0

∞

[g (r ) − 1]r 2

sin(qr ) dr qr

(16)

When the simple averaged polydispersity is taken into account, the structure factor can be calculated using eq 17. S av(q) =

∑ xαSα(q) (17)

α

V (r ) = βυ(r ) =

av

Here, S (q) is the averaged structure factor, Sα(q) is the monodisperse structure factor of species α, and xα is the molar fraction of the species α. This approach does not have any influence on the structure factor at zero scattering vector S(0) and the calculation itself can be easily performed in a short time and any additional information on the form factor is not required. The effective structure factor for polydisperse hard spheres depends on both the scattering properties of the single particles and the intraparticle interactions and is given in eq 18.9,12−14 I(q) = P(q)S eff (q) → S eff (q) =

α ,β=1

fα fβ Bα (q)Bβ (q)Sαβ(q)

z 2e02 e−κ(a − 2a) · 4πεε0kT r(1 + κa)2

(20)

where z is the number of unit electron charges on the particle, e0 is the unit charge of an electron, ε0 is the permittivity of vacuum, ε the relative dielectric constant, k the Boltzmann constant, T the temperature, κ the Debye screening parameter, a the radius of a sphere, and r the center to center distance of the spheres. This method leads to stable pair distance distribution functions, but to get the correct structure factor, only two of the three parameters (radius, charge, volume fraction) are varied due to the linear dependence of the results.7 This model is fast but still inaccurate (eq 21).

1 P(q)

c(r ) = −βυ(r )

m

∑

(19)

(21)

GIFT6: Charged Spheres, Monodisperse (Yukawa Potential, HNC Closure). Here, the model makes use of the hyper-

(18) 3743

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netted-chain (HNC) closure relation, eq 22.15 This method is a compromise between the exact but slow RY (Rogers Young relation) and the fast but not exact RMSA (rescaled mean sphere approximation) calculations.

g (r ) = e−βυ(r) + b(r) − c(r)

known as a technique, whose results are strongly influenced by the presence of large particles. This can be explained by the fact that the intensity is directly proportional to the hydrodynamic radius of the particles. As a consequence, the few large particles exhibit a higher total scattering contribution than many small particles. In contrast, for the number weighted DLS results the intensity is directly proportional to the radius to the power of six. Here, it can be seen that the influence of large particles diminishes and that the scattering contribution of the small scattering centers determines the mean particle size. The small-angle X-ray scattering experiments of the aqueous nanocrystalline cellulose suspensions (aNCS) have been performed using different concentrations, but only for the highest concentrations (1.0 wt %) could a sufficient scattering contrast be obtained. In Figure 2, the experimental scattering curve after the background-, water-, and capillary-scattering subtraction and binning is shown. The scattering curve shows a strong decay of the scattering intensity. For the calculation of the radius of gyration via the Guinier approximation, the experimental scattering curve (Figure 2) was plotted on one hand as Guinier-plot (plotting ln(I(q)) vs q2 whereby the slope is proportional to Rg2) and on the other hand as cross section Guinier-plot (plotting log(I(q)*q) vs q2 whereby the slope is proportional to Rc2). The results of both fits for different qranges are summarized in Table 2. It can be clearly seen that the radii of gyration depend on the selected q2-region as well as on the chosen geometry of the particles. For globular particles, the Guinier plot yields Rg values between 9.2 and 12.2 nm. However, the aNCS are rod-like particles; therefore, another extrapolation approach for the calculation of the radius of gyration, the cross section Guinier-plot, has been used. The obtained Rc values range from 4.3 and 8.5 nm depending on the range and more detailed information is listed in Table 2. Although we have a different cellulose source, the values of Rc are in a good agreement with the literature where radii of gyration in the range between 5 nm and 7 nm have been determined for nanocrystals derived from cotton.28 Nevertheless, this example also shows that the applied range for a fitting of the radius of gyration has a significant influence on the final results. A possible alternative to these procedures is the use of GIFT.7−11 The advantages comprise a simultaneous determination of the structure and form factor, the applicability in highly concentrated suspensions/solutions, and the possibility to include charges of the particles in the fitting procedure. In the first step of the analysis, the scattering curve was subjected to a fitting procedure using the GIFT software to obtain the real space information on one hand via the indirect Fourier transformations (IFT) to extract the form factor P(q), and on the other hand via the generalized indirect Fourier transformation (GIFT) which allows to separate not only the form factor P(q), but also simultaneously the structure factor S(q) according to different model approaches, which were described in the Theoretical section. For this purpose, we employed a spherical approach (SA) and a rod-like approach (RA), which models the cross section of a cylinder. Although the nanocrystals are clearly elongated, we are interested in whether a spherical approach can be used for the description of the aNCS which of course requires a certain degree of flexibility of the models during the fitting procedure. Such a flexibility can then result in a reasonable description of a rod-like particle even when a spherical approach is used for modeling. In principle, GIFT1 lacks such a flexibility due to the hard sphere

(22)

This structure factor model is reliable and leads to stable p(r) functions. GIFT7: Charged Spheres, Monodisperse (Yukawa Potential, RY Closure). The model calculations here are based on the Rogers Young (RY) closure relation, eq 23.15,27 This method is very accurate, precise, and thermodynamically consistent, but requires more computing time than the aforementioned models ⎞ ⎛ 1 g (r ) = e−βυ(r)⎜1 + [e f (r)(b(r) − c(r)) − 1]⎟ f (r ) ⎠ ⎝

■

f (r ) = 1 − e − α r

with (23)

RESULTS AND DISCUSSION The nanocrystalline cellulose suspensions are prepared by treatment of microcrystalline cellulose with sulfuric acid followed by purification. Besides the preferred hydrolysis of the amorphous parts of the microcrystalline cellulose, a reaction of sulfuric acid at the NCC surface takes place, which results in the formation of sulfate esters. As a consequence, the nanocrystals exhibit a high negative surface charge which can be easily analyzed using zeta potential determination. The zeta potential is a very good indicator of the stability of particles in suspensions. According to the DLVO theory, particles are more stable the higher the absolute value of the ζ-potential is. In the case of the aNCS, the zeta potential is −96 ± 2 mV indicating high stability at native pH. The total sulfur content of our nanocrystals has been determined by ICP-MS and confirms the results from zeta potential determination; the sulfur content was determined to be 5650 ± 10 mg/kg. The shape and morphology of the crystals is in agreement with other reports on nanocrystalline cellulose where sulfuric acid has been used as hydrolyzing agent. On the AFM image (Figure 1, left), rod-

Figure 1. Left: AFM topography image (1 μm × 1 μm) of aNCS deposited on a PEI coated on silicon wafer. Right: Comparison of DLS results on mean particle sizes using number and intensity weighted size distributions.

like particles are present which show a full coverage of the surface. The width of these particles is below 50 nm, while the length is in the range of a few hundred nanometers. Similar results have been found using dynamic light scattering measurements (Figure 1, right). Here, the multimodal size distributions of the intensity as well as the number weighted results are shown. The intensity weighted DLS is 3744

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Figure 2. SAXS curve of the aNCS after background correction and water-capillary subtraction at absolute scale.

Table 2. Radius of Gyration Calculated via Guinier Approximation in Dependence of Chosen Q Ranges and Assumed Geometry (Spheres Vs Rods) type globular globular rod-like rod-like

q(min)2 [nm−2]

equation

I(q) = I(0)e

q2Rg 2 /3

Ic(q) = Ic(0)eq

2

Rc 2 /2

2.46 2.46 6.08 2.46

× × × ×

q(max)2 [nm−2]

−3

10 10−3 10−3 10−3

9.29 1.13 9.29 1.13

approach used within the fitting procedure. Nevertheless, it is used in this study to get an idea how large the deviations are when improper models are employed. The approximated curves of the SA (left) and RA (right) using different models are compared in Figure 3 (upper row) and the corresponding structure factors are shown in Figure 3 (lower row). Without any quantification of the results, it can be clearly seen that the GIFT calculations differ from the IFT calculation, in particular, at low values of q. In principle, repulsive interactions between the particles are accompanied with a decrease in the fitted intensity at low scattering angles for the GIFT approach in comparison to an IFT treatment. The GIFT approach enables an estimate of the contribution of S(q) to the total scattering intensity and indicates the presence of repulsive interactions between the individual particles due to a decrease of the fitted intensity, which is consistent with the ζ-potential measurements. However, there are some differences between the results obtained by the SA and RA. While in the SA models, all approximated curves are decreasing in the fitted q range, for the RA this is not the case. In contrast, for some treatments (IFT, GIFT1, 2, 4, 5) a nearly linear increase in intensity is observed. This is a sign that these models underestimate interaction forces between the individual nanocrystals. As a consequence, these GIFT models also exhibit much smaller structure factors than the ones with a better description of the real situation (GIFT3, 6, 7). Besides the estimation of attractive/repulsive forces, the pair distance distribution function (PDDF) which is obtained by indirect Fourier transformation of I(q) has a slightly different

× × × ×

10−3 10−3 10−3 10−3

radius of gyration [nm]

± [nm]

9.2 12.2 4.3 8.5

0.5 2.1 0.3 1.7

shape when applying the GIFT procedure (Figure 4). The integrals of the p(r) functions are proportional to the number of distances that can be found inside the particle with the same electron density (within the interval r and r + dr). The characteristic shape of all p(r) functions indicates the rod-like shape of the nanocrystals which was proposed by Orts et al.29 The oscillations of the p(r) function in Figure 4 above a radius of around 15 nm indicate the staggered nature of the cellulose nanowhiskers and the axial inhomogeneities which originate from the quasi-periodical changes in the electron density along the cylinder axis. These oscillations are even more pronounced for the cross section approach (Figure 4 lower row). This can be explained by the 20 cubic B-splines which were used in both cases to obtain the p(r) and pc(r) functions. These oscillations are also reflected in the error bars and increase with increasing radius. The maxima of the pc(r) functions of the cross section of the CNC rods (Figure 4 lower row) is around 2.4 nm. In Figure 5, the schematic view of a threaded cellulose nanocrystallite according to the GIFT results of the spherical and cross section approach is shown. Using these data, the distance between the cellulose chains within the nanocrystals as well as the shape of those can be calculated and assumed. For this sample, the data treatment yields in general a larger radius of gyration with standard IFT type calculations (Rg = 14.5 nm; Rc = 5.2 nm) than the GIFT procedure (Rg = 13.4−14.0 nm; Rc = 4.1−5.1 nm) which originates from the extraction of S(q) from I(q). Although absolute differences are quite small (ΔRg: 0.5−1.7 nm, ΔRc: 0.1−1.1 nm), the relative ones are remarkable and cover a range from 4% to 8% (ΔRg) and 0.2% to 20% (ΔRc). It must be 3745

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Figure 3. Approximated scattering curves following the spherical (upper row, left) and rod-like approach (upper row, right) and corresponding structure factors (lower row).

noted that the values, both Rg and Rc, obtained by the GIFT procedure are much larger than those obtained by the Guinier approximation. The differences are based on the method of determination. While the Guinier approximation uses eqs 10 and 12, respectively, the GIFT program uses the pair distance distribution function for the elucidation of Rg according to eqs 11 and 13. However, in our opinion the determination of radii of gyration of rod-like particles using RA in concentrated aqueous suspensions can only be described by the pair distance distribution rather than the Guinier approximation where the intensity drops to zero in the Guinier range.18 Further, when the RA is applied, the cross section results of the pc(r) yield similar Rc values (see Table 3) compared to the cross section Guinier results (Rc = 4.3 nm for (6.08 × 10−3) < q < (9.29 × 10−3 nm−1). In addition, the GIFT approach enables an estimation of the radius of the aNCS in aqueous media. As depicted in Table 3, the obtained radii are between 38 and 41 nm (SA), which is in excellent agreement with AFM data as shown in Figure 1, left. The radii of the cross section are between 18 and 20 nm, which is also in a good agreement with the AFM data. In addition, some models (GIFT models 2 to 4) are able to estimate the polydispersity of the particles. According to these models, PDI are in a range from 0.11 to 0.15 for the SA, while for the RA 0.15 to 0.34 have been determined. The results of the spherical

approach are consistent with data obtained from dynamic light scattering where PDI of 0.18 has been determined, but the RA results are significantly different which is probably due to a larger deviation of the width of the particles (cross section) rather than the diameter. The GIFT models 5 to 7 are taking into account the effective charge. GIFT5 performs very poorly in the determination of S(q), therefore it is less surprising that the effective charge also is not predicted in a reasonable manner. In comparison to GIFT6 and 7, which yield Zeff of 16 and 14 for RA, respectively, GIFT5 overestimates the charge with Zeff = 44. In principle, Zeff should be in a similar region for the SA and the RA (in fact, Zeff should be slightly larger for the SA in comparison to the RA); therefore, the larger the deviation between Zeff (SA) and Zeff (RA), the less accurately the model describes the system. In this context, the smallest deviation ΔZeff (SA-RA) in the effective charge is present for GIFT7 and the largest for GIFT5. For the sake of comparison, it would be desirable to directly access this effective charge by experimental means. In contrast to the bare charge of a colloidal particle, which can be easily determined by various methods (e.g., titration), the precise determination of effective charges is more tricky to handle and requires a laborious theoretical treatment of experimentally derived data obtained by, e.g., zeta potential determinations through various renormalization approaches. In the end, one may end up with 3746

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Figure 4. Pair distance distribution functions PDDF, p(r) and pc(r), obtained by different model approaches.

Table 3. Summary of Results Obtained by the Different Model Approaches Using SA and RA with r Being the Radius, Rg the Radius of Gyration, Rc the Radius of Gyration of the Cross Section, Vf the Volume Fraction, and PDI the Polydispersity index Figure 5. Schematic view of a threaded cellulose nanocrystallite according to the GIFT results.

results that are highly dependent on the used theoretical approaches making a clear judgment quite difficult. As these findings show, some caution is necessary when interpreting the results, and always at least one additional technique should be used for validation purposes. As discussed in the Theoretical sections, GIFT models 1 to 4 do not consider the charge of the particles to a full extent. Therefore, a similarity of the results from GIFT1 to GIFT4 can be based on error cancellation yielding acceptable results. This can be explicitly seen in the calculated volume fractions, which differ substantially from model GIFT1 to GIFT4 and GIFT5, respectively. In our opinion, GIFT6 and 7 are the most reliable models for charged particles, and additionally, the effective charges per particle can be calculated. However, more data (more polysaccharide nanocrystals samples) must be obtained in order to make more precise statements on the reliability of the obtained charges via GIFT.

a

3747

SA

r [nm]

Rg [nm]

Vf

PDI

effective charge

IFTa GIFT1 GIFT2 GIFT3 GIFT4 GIFT5 GIFT6 GIFT7 RA

37.9 41.4 39.7 39.7 40.5 33.7 45.9 r [nm]

14.5 14.0 13.4 13.5 13.5 12.8 13.7 13.9 Rc [nm]

0.048 0.116 0.112 0.104 0.100b 0.100b 0.094b Vf

0.146 0.112 0.109 PDI

28.5 26.6 10.9 effective charge

IFTa GIFT1 GIFT2 GIFT3 GIFT4 GIFT5 GIFT6 GIFT7

20.0 18.1 20.0 17.5 20.0b 19.0 17.5

5.2 4.9 5.1 4.1 4.3 4.9 4.6 4.9

0.028 0.010 0.239 0.123 0.027 0.160b 0.160b

0.151 0.286 0.336 -

43.6 16.2 13.8

No model assumption. bConstant parameter during fitting.

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transformation under consideration of the effective structure factor for polydisperse systems. J. Appl. Crystallogr. 1999, 32, 197−209. (10) Bergmann, A.; Fritz, G.; Glatter, O. Solving the generalized indirect Fourier transformation (GIFT) by Boltzmann simplex simulated annealing (BSSA). J. Appl. Crystallogr. 2000, 33, 1212−1216. (11) Bergmann, A.; Brunner-Popela, J.; Fritz-Popovski, G.; Früwirth, T.; Glatter, O.; Innerlohinger, J.; Mittelbach, R.; Weyerich, B. PCG Software Package version 3.01.10; Graz, 1999−2009. (12) Vrij, A. Mixtures of Hard-Spheres in the Percus-Yevick Approximation - Light-Scattering at Finite Angles. J. Chem. Phys. 1979, 71, 3267−3270. (13) Blum, L.; Stell, G. Polydisperse Systems. 1. Scattering Function for Polydisperse Fluids of Hard or Permeable Spheres. J. Chem. Phys. 1979, 71, 42−46. (14) Blum, L. Correction. J. Chem. Phys. 1980, 72, 2212−2212. (15) Klein, R.; D’Aguanno, B. Light Scattering. Principles and Development. Chapter: Static scattering properties of colloidal suspensions; Oxford University Press: New York, 1996; pp 30−102. (16) Glatter, O. New Method for Evaluation of Small-Angle Scattering Data. J. Appl. Crystallogr. 1977, 10, 415−421. (17) Glatter, O. Interpretation of Real-Space Information from Small-Angle Scattering Experiments. J. Appl. Crystallogr. 1979, 12, 166−175. (18) Glatter, O.; Kratky, O. Small Angle X-ray Scattering; Academic Press Inc. Ltd: London, 1982. (19) Guinier, A.; Foumet, G. Small-Angle Scattering of X-rays; John Wiley: New York, 1955. (20) Ailawadi, N. K. Equilibrium Theories of Simple Liquids. Phys. Rep. 1980, 57, 241−306. (21) Hayter, J. B.; Penfold, J. An Analytic Structure Factor for Macroion Solutions. Mol. Phys. 1981, 42, 109−118. (22) Kaler, E. W. Small-Angle Scattering from Complex Fluids, In Modern Aspects of Small-Angle Scattering; Kluwer Academic Publishers, Dordrecht, 1995. (23) Glatter, O. Computation of Distance Distribution-Functions and Scattering Functions of Models for Small-Angle Scattering Experiments. Acta. Phys. Austriaca 1980, 52, 243−256. (24) Glatter, O. Evaluation of Small-Angle Scattering Data from Lamellar and Cylindrical Particles by the Indirect Transformation Method. J. Appl. Crystallogr. 1980, 13, 577−584. (25) Percus, J. K.; Yevick, G. J. Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 1958, 110, 1− 13. (26) Hansen, J. P.; Hayter, J. B. A Rescaled Msa Structure Factor for Dilute Charged Colloidal Dispersions. Mol. Phys. 1982, 46, 651−656. (27) Rogers, F. J.; Young, D. A. New, Thermodynamically Consistent, Integral-Equation for Simple Fluids. Phys. Rev. A 1984, 30, 999−1007. (28) Elazzouzi-Hafraoui, S.; Nishiyama, Y.; Putaux, J. L.; Heux, L.; Dubreuil, F.; Rochas, C. The shape and size distribution of crystalline nanoparticles prepared by acid hydrolysis of native cellulose. Biomacromolecules 2008, 9, 57−65. (29) Orts, W. J.; Godbout, L.; Marchessault, R. H.; Revol, J. F. Enhanced ordering of liquid crystalline suspensions of cellulose microfibrils: A small angle neutron scattering study. Macromolecules 1998, 31, 5717−5725.

CONCLUSION The use of the generalized indirect Fourier transformation GIFT allows characterizing concentrated and interacting systems with small-angle X-ray scattering in laboratory scale. We demonstrated at the example of aNCS that this approach is particularly advantageous for analyzing polysaccharide samples in comparison to the standard IFT treatment. The estimated radii of gyration Rg and Rc seem to be overestimated by IFT and all models using GIFT revealed smaller values. Additionally, the use of GIFT during refinement and fitting allows the use of highly concentrated solutions as well as suspensions, where particle−particle interactions play an important role. Such samples can be hardly analyzed with other approaches such as the widespread IFT approach, where particle−particle interactions must be excluded. Moreover, the polydispersity as well as the charge of the particles can be calculated.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. ‡ H.M.A.E., S.S., A.D., T.M., V.R., M.S.-S., and K.S.-K. are members of the European Polysaccharides Network of Excellence − EPNOE.

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ACKNOWLEDGMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme [FP7/2007-2013] under grant agreement no. 214015. The Slovenian research agency (ARRS) is gratefully acknowledged for financial support (project no. L2-4101). The authors thank Prof. Otto Glatter from the University of Graz for all the fruitful discussions about the GIFT method and SAXS measurements in general.

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REFERENCES

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