We discuss the structural and dynamical properties of a colloidal glass of silica nanoparticles in a water-lutidine mixture probed using photon correlation techniques. We describe the small angle set-up used to perform X-ray photon correlation spectroscopy (XPCS) experiments and the procedure followed to measure the volume fraction of the sample. We describe the structure of the glass using a short range potential model and a theoretical structure factor within the mean spherical approximation. The dynamics finally is characterized by a Gaussian-like intermediate scattering function which is not compatible with the classical picture of an heterogeneous diffusive process

Dallari, F.

Grübel, G.

Martinelli, A.

Monaco, G.

Zontone, F.

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DOI 10.1393/ncc/i2018-18104-xCommunications: SIF Congress 2017IL NUOVO CIMENTO 41 C (2018) 104The glass transition in colloidal suspensions of silica nanoparticlesin a water-lutidine mixture: A photon correlation studyA. Martinelli(1) on behalf of F. Dallari(1), G. Grübel(2), F. Zontone(3) andG. Monaco(1)(1) Department of Physics, University of Trento - Trento, Italy(2) Department of Physics, University of Hamburg - Hamburg, Germany(3) European Synchrotron Radiation Facility - Grenoble, Francereceived 30 January 2018Summary. — We discuss the structural and dynamical properties of a colloidalglass of silica nanoparticles in a water-lutidine mixture probed using photon corre-lation techniques. We describe the small angle set-up used to perform X-ray photoncorrelation spectroscopy (XPCS) experiments and the procedure followed to mea-sure the volume fraction of the sample. We describe the structure of the glass usinga short range potential model and a theoretical structure factor within the meanspherical approximation. The dynamics finally is characterized by a Gaussian-likeintermediate scattering function which is not compatible with the classical pictureof an heterogeneous diffusive process.1. – IntroductionColloidal systems have been investigated deeply in the last few decades. Nowadaysnanoparticles (NP) are arousing interest due to the wealth of possibilities that theyare manifesting. Gold and synthetic NP for gene and drug delivery, noble metals NPfunctionalized with DNA to build complex nanostructures, quantum dots and catalistsare only few examples [1-3].In this article we discuss the structural and dynamical properties of a concentratedsuspension (volume fraction φ = (42±2)%) of silica nanoparticles in a water-2,6 lutidine((CH3)3C5H3N) mixture (0.25 lutidine mass fraction). The water-lutidine solution hasa coexistence curve of the two liquid phases with a lower critical point near the massfraction ∼ 0.3 of lutidine in water and a critical temperature of 34 ◦C [4]. This binarymixture exhibits an uncommon feature when it is placed in contact with silica surfaces:a so-called wetting transition takes place, that is the lutidine separates from water andcovers silica surfaces with a layer thickness that depends on the temperature of thesystem [5]. This layer weakens the repulsive electrostatic interaction between the colloidsCreative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 A. MARTINELLI et al.and induces aggregation in diluted solutions of silica nanoparticles in near critical water-lutidine mixtures [6, 7].It was observed using photon correlation techniques that high volume fraction sus-pensions do not show a temperature induced aggregation but a glass-glass transitionbetween a repulsive and an attractive glass [8].We report here a detailed study of a colloidal repulsive glass of 100 nm diameter SiO2particles in a water-2,6 lutidine mixture (0.25 lutidine mass fraction) at the packingfraction φ = (42 ± 2)% and at the temperature T = (32.00 ± 0.01) ◦C.2. – Experimental details2.1. Photon correlation. – Photon correlation is a well-known technique [9]. In the so-called homodyne configuration one performs the autocorrelation of the scattered intensityat a given scattering vector q:(1) g2(q, τ) :=〈I(q, t)I(q, t + τ)〉t〈I(q, t)〉t2= 1 + α|F (q, τ)|2,where 〈...〉t is a temporal average, α is the contrast and F (q, τ) is the intermediatescattering function. The last equality is known as the Siegert relation, valid in theGaussian approximation [9]. Since the intermediate scattering function is the Fouriertransform of the Van-Hove self–space-time correlation function, a photon correlationexperiment gives direct information on the dynamical properties. In most of the cases,the intermediate scattering function can be modelled as a modified exponential function(KWW function) [10]:(2) F (q, τ) ∝ e(−Γτ)β ,where β is known as stretching parameter and Γ−1 is the decay time.It is clear that in order to obtain a reliable autocorrelation function, one needs tointegrate for a time much longer than the typical decorrelation time of the intermediatescattering function. However, in glassy-like systems this time could exceed hundredsof seconds, causing difficulties in the use of this technique. To overcome the relatedstatistical limitations the so-called multispeckle approach has been developed [11]. Withthe use of a 2D detector, one can perform an average over the pixels related to the sameexchanged wave vector q, reducing the needed integration time of a factor equal to thenumber of speckles which are taken into the average.2.2. Experimental set-up. – We carried out an X-Ray photon correlation experimentin small angle geometry (XPCS-SAXS) at the coherent station of the ID10 beamline atthe European Synchrotron Radiation Facility (ESRF) in Grenoble (France).Synchrotrons are basically incoherent sources of X-rays. The way to get a partiallycoherent beam is to use slits, pinholes and monochromators. Indeed transverse andlongitudinal coherence lengths at a distance R from the source can be defined as [12]:(3) ξt λR2Σ, ξl λ2Δλ,where λ and Δλλ are, respectively, the wavelength and the relative bandwidth of the beamand Σ is the source size.COLLOIDAL SUSPENSIONS OF SILICA NANOPARTICLES IN A WATER-LUTIDINE MIXTURE 3Fig. 1. – Scheme of the experimental set-up for small angle coherent X-ray scattering. Thescattered intensity is collected with the Maxipix (2D single-photon counterdetector) [14].With a Si(111) monochromator one is able to achieve an energy resolution Δλλ =1.4×10−4, which corresponds to ξl  1 μm at E = 8.1 keV. With regard to the transversecoherence, at a given energy the decisive factor is the ratio RΣ . For this reason, thesample is kept far (R ∼ 50m) from the source and with a combination of lenses and slitsa coherent beam of 10 × 10 μm2 size is obtained [13].In fig. 1 the scheme of the experimental set-up is reported. In order to perform acoherent scattering experiment the sample volume must be illuminated coherently. Forthis reason, the path-length difference (PLD) for rays in the scattering volume cannotexceed the longitudinal coherence ξl. The PLD can be written as [12](4) PLD  2W sin2 θ2+ d sin θ,where W is the thickness of the sample, θ is the scattering angle as reported in fig. 1 andd is the beam diameter. A sample thickness W = 500 μm was chosen for the experiment,which fulfils the requirements of eq. (4) for the whole detector surface.It can be shown [9] that the maximum contrast in the autocorrelation function isachieved when at most one speckle impinges on the detector. Since the multispeckletechnique is used to increase the statistical accuracy, the best performance is obtainedwith speckles of the same size as the pixel (55 × 55 μm2). The speckle linear size can beestimated as(5) lc ∼λΩ 1.22 · λDd,where λ is the wavelength of the incident beam, Ω is the angle of the source subtended atthe detector, D is the source-detector distance, d is the beam diameter and the term 1.22is introduced to take into account the cylindrical symmetry of the scattering volume. Adistance D = (5.22 ± 0.01)m was used in the experiment, which implies speckles with alinear size of ∼ 100 μm.4 A. MARTINELLI et al.2.3. Sample preparation. – The samples were prepared from a diluted suspensionof silica Sicastar R©(Micromod) nanoparticles (100 nm diameter, mass concentration100 mg/mL of solution). The right amount of 2,6-lutidine (Sigma-Aldrich R©) was addedto the solution to reach a lutidine-water mass fraction Clw = 0.25. The obtained sus-pension was then centrifuged and the exceeding solvent removed to obtain the desiredvolume fraction of silica with respect to the solution. The sample was then shaken ona vortex mixer to get a homogeneous suspension and filled into Mark-tubes capillariesmade of borosilicate glass (Hilgenberg R©) of outer diameter 0.5 mm and wall thickness0.01 mm and placed in a ultrasonic bath for 1 minute to remove air bubbles.3. – Measuring the volume fractionIn order to obtain high volume fractions (φ ∼ 40%) most of the solvent has to beremoved. Since the particles diffuse into the solution after centrifugation, it is difficultto estimate the exact final volume fraction of the colloidal system. It is then worth toperform a direct measurement of the volume fraction.The transmitted part of an X-ray beam through a sample of thickness x can bemodelled as [15](6) I = T · I0 = I0e−μx,where I and I0 are, respectively, the transmitted and incident beams, T is the transmis-sion coefficient and μ is the attenuation coefficient of the sample. In the case of a colloidalsuspension, one can model it with μ = (1−φ)μs +φμp, where φ is the volume fraction, μsis the attenuation coefficient of the solvent and μp is the one relative to the particles. Totake into account the thickness of the walls of the capillary, the transmission coefficient Tis divided by the one obtained for an empty borosilicate capillary (Tempty = 0.83±0.02).Finally, the volume fraction can be calculated as(7) φ =− log( TTempty )1ξ − (CL · μL + (1 − CL) · μW )μSiO2 − (CL · μL + (1 − CL) · μW ),where CL is the mass fraction of lutidine in the solvent, μL and μW are the absorptioncoefficients for lutidine and water, respectively, and ξ is the inner diameter of the capillary.An intensity profile (fig. 2) is obtained scanning the sample across the beam whilemeasuring the intensity on the 2D detector. The points outer of the capillary are fittedwith a linear law while the bottom of the capillary is fitted with a third power polynomialfunction.4. – Structural propertiesThe experimental scattered intensity is extracted from the images with the followingprocedure. For each image, dead pixels are masked and the intensity integrated az-imuthally (ϕ) over rings of width Δr centred on the transmitted beam (fig. 1). Then 100of these spectra (corrected for the subtended solid angle) are averaged to get a standarddeviation for each point I(q).In the kinetic approximation the scattered intensity is(8) I(q ) ∝ |f(q )|2S(q ),COLLOIDAL SUSPENSIONS OF SILICA NANOPARTICLES IN A WATER-LUTIDINE MIXTURE 5Fig. 2. – Example of intensity profile obtained for a volume fraction of φ = (42 ± 2)%. Toget a reliable value of I0 the data corresponding to the transmission out of the sample () arefitted with a linear law; those corresponding to the bottom of the capillary (×) are fitted witha third power polynomial function. To get a correct estimate of the diameter of the capillary,the half-maximum width is extracted using a linear interpolation.where |f(q )|2 is the form factor and S(q ) is the structure factor.4.1. Form factor . – The form factor |f(q )|2 is nothing but the Fourier transform ofthe charge density. In the case of colloidal suspensions the form factor is described interms of the relative density ρ = ρp −ρs, where ρp is the charge density of the suspendedparticles and ρs is the charge density of the solvent.The form factor of a sphere is [16](9) |f(|q |)|2 =∣∣∣∣ρ43πR3[sin(qR) − qR cos(qR)](qR)3∣∣∣∣2,where R is the radius of the sphere and q the exchanged wave vector. It is important tostress the fact that for symmetry reasons, |f(q )|2 = |f(|q |)|2.Since the particles are not monodispersed one has to consider the contribution in theform factor due to different particle’s radii. The particle size distribution is modelledwith the Schultz distribution(10) fz(R) =1z![z + 1/r̄]z+1rz exp−(z + 1)r/r̄with r̄ the average particle radius and z a parameter that is related to the second momentof the distribution, i.e. 〈R2〉 − 〈R〉2 = r̄2/(z + 1). For this particular distribution an6 A. MARTINELLI et al.Fig. 3. – (a) Experimental form factor after background subtraction (◦) and best fit (dashedline) performed using a Levenberg-Marquardt nonlinear least-squares algorithm. The obtainedparameters are 〈R〉 = (51±1) nm andp〈R2〉 − 〈R〉2/〈R〉 = (8±1)% (b) Background measuredwith a capillary and the solvent.analytic function exists to model the form factor [17].In fig. 3 the experimental form factor is reported after background subtraction. Thefitting function is the form factor for polydispersed spheres convoluted with the width ofthe rings used for the calculation of the experimental intensity.4.2. Structure factor . – The structure factor is strictly related to the correlation func-tion between particles. Indeed it comes out in the scattered intensity as the interferenceof the scattered field from different particles. Thanks to this, one can recover structuralinformation such as the pair correlation function (g(r)) and the interaction potential. Acomplete treatment can be found in dedicated books [18] and here only few fundamentalresults are reported.The structure factor for an isotropic homogeneous system can be written as(11) S(q) − 1 = n0∫dr4πr2sin qrqr[h(r)]with n0 the average particle density and h(r) the total correlation function functiondefined as g(r) − 1. The Ornstein-Zernike equation [19] expresses the total correlationfunction in terms of the direct correlation function ĉ(q):(12) ĥ(q) =ĉ(q)1 − ρĉ(q) .COLLOIDAL SUSPENSIONS OF SILICA NANOPARTICLES IN A WATER-LUTIDINE MIXTURE 7Fig. 4. – Experimental structure factor (◦) and the best fit (dashed line) with eq. (14).The function ĉ(q) is the correlation between particles 1 and 2 only due to the directinteraction between the two particles. Following the idea of the DLVO theory [20,21], theinteraction potential between the colloids is modelled with a long-range van der Waalsattraction (neglected in the present case) and a short-range electrostatic repulsion:(13) U(r) =⎧⎨⎩∞, if r < σ,πε0ε σ2Ψ02e−k(r−σ)r, if r > σ,with σ the diameter of the particles, r the relative distance, Ψ0 the surface potential andk the inverse of the screening length.It is possible to solve analytically eq. (12) within the mean spherical approxima-tion [22,23]:(14)⎧⎨⎩h(x) = −1, if x < 1,c(x) = −βU(x) = −γ e−k′xx, if x > 1,where x = r/σ and γ = βπεε0σΨ02ek with β the Boltzmann factor 1/kBT . The to-tal dielectric constant ε is calculated linearly combining the ones relative to water andlutidine.In fig. 4 the experimental structure factor is reported together with the best fit.The parameters of the fitting function [23] are the volume fraction φ, the inverse of thereduced screening length k′ = kσ and the surface potential Ψ0. The obtained parameters8 A. MARTINELLI et al.Fig. 5. – Autocorrelation function (◦) calculated on the peak of the structure factor (q =0.07 nm−1) and the best fit (dashed line) using eq. (1) and eq. (2). The obtained parametersare Γ−1 = (72 ± 1) s and β = (2.0 ± 0.1).with a non-linear fit (Levenberg-Marquardt least-squares algorithm) are k′ = 2.5 ± 0.1,Ψ0 = (15.7 ± 0.5)mV and φ = 0.41 ± 0.01. This last value is in agreement with the oneobtained with the transmission measurements (φ = 0.42 ± 0.02).5. – Dynamical propertiesIt is well established that near the glass transition the stretching parameter (β) of theKWW function is usually smaller than 1. This behaviour is attributed to a distributionof simple exponential relaxation processes [24,25].In fig. 5 the autocorrelation function is reported, calculated at the peak of the struc-ture factor (q = 0.07 nm−1) with the multispeckle approach. The value obtained fromthe fit for the stretching parameter is β = (2.0 ± 0.1). A compressed (β > 1) behaviourcannot be explained with a superposition of simple exponentials. This peculiar decayis observed in colloidal systems [26] and in metal glasses [27] where it is attributed tothe presence of internal stresses. Simulations on colloidal gels have recently shown thatthe intermittent microscopic dynamics (rearrangement of the gel connections) leads to afaster than exponential relaxation process and a super-diffusive particle dynamics [28,29].A clear understanding of this phenomenon remains however a topic of current research.COLLOIDAL SUSPENSIONS OF SILICA NANOPARTICLES IN A WATER-LUTIDINE MIXTURE 96. – ConclusionsWe have characterized a colloidal glass of silica nanoparticles in a water-lutidine by-nary mixture. We have modelled the interaction potential with a short-range screenedCoulombic repulsion. The calculated structure factor in the mean spherical approxima-tion is in agreement with the experimental data. From a dynamical point of view, wehave observed a compressed behaviour of the intermediate scattering function. Despitethis feature is common in soft matter systems, the process beyond it is still argument ofdebate and it is not fully clarified.∗ ∗ ∗We acknowledge the European Synchrotron Radiation Facility for provision of syn-chrotron radiation facilities, in particular we thank Karim Lhoste for the technical sup-port and Diego Pontoni for the assistance in using the Soft Matter Laboratory of PSCM.REFERENCES[1] Samanta A. and Medintz I. L., Nanoscale, 8 (2016) 9037.[2] Nozik A. J., Beard M. C., Luther J. M., Law M., Ellingson R. J. and Johnson J.C., Chem. Rev., 110 (2010) 6873.[3] Turner M., Golovko V., Vaughan O., Abdulkin P., Berenguer-Murcia A.,Tikhov M., Johnson B. and Lambert R., Nature, 454 (2008) 981.[4] Handschy M. A., Mockler R. C. and O’Sullivan W. J., Chem. Phys. Lett., 76 (1980)172.[5] Beysens D. and Estéve D., Phys. Rev. Lett., 54 (1985) 2123.[6] Beysens D. and Narayanan T., J. Stat. Phys., 95 (1999) 997.[7] Pontoni D., Narayanan T., Petit J.-M., Grbel G. and Beysens D., Phys. 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The glass transition in colloidal suspensions of silica nanoparticles in a water-lutidine mixture: A photon correlation study

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The glass transition in colloidal suspensions of silica nanoparticles in a water-lutidine mixture: A photon correlation study

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