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II. We then present our new MALS appa- electronics. Ferri and coworkers report a setup covering ratus in Sec. III, before describing its angular calibration scattering angles up to 15deg5,6, using a commercially in Sec.V. A series of tests of the apparatus’ performances available cell but again a custom photodiode array with for both SLS and DLS are presented in Sec. VI, before dedicated electronics. A different approach is used in the the concluding remarks of Sec. VII. commercially available apparatus of Ref.9, which covers scattering angles from a fraction of degree up to about 40 deg by varying the propagation direction of the inci- dent beam. Accordingly, several distinct measurements are required to sample the full angular range, each mea- surement covering about 5 deg. Chou and Hong10 report a CCD-based setup in the range 2deg <θ< 25 deg using II. POPULAR OPTICAL LAYOUTS FOR SALS AND a scheme originally proposed by Ferri for SALS3. Unfor- USALS tunately, however, no detailed description and charac- terization of the apparatus performances are provided. Finally, one may take advantage of the good-quality op- Most MALS setups are based on the same optical lay- out as that for SALS and USALS, examples of which are tics of modern microscopes to build an apparatus that combines imaging with low- or mid-angle scattering. Ka- shown in Fig. 1. In the top scheme, Fig. 1a, adopted in Refs.4,9,14,15, the scattered intensity is measured in the plan et al.11 report a DLS apparatus based on an in- focal plane Σ of a so-called Fourier lens of focal length verted microscope that covers scattering angles between 20.6deg and 55.1 deg, while Celli et al.12 demonstrate f. Neglecting refractions at the solvent-cell and cell-air interfaces, a point on Σ at a distance r from the optical DLS in a upright microscope for θ ≤ 12 deg. In both −1 cases, measurements are performed at one single angle axis corresponds to a scattering angle θ = arctan(rf ). In order to avoid artifacts due to light leaking from at a time. In Ref.13, a microscope- and CCD-based static light scattering instrument is presented, covering the intense transmitted beam, a hole may be drilled in the detector to let the transmitted beam pass through the range 0.9 µm ≤ q ≤ 18 µm, corresponding approxi- 4–6,14,15 mately to 3.3deg ≤ θ ≤ 70 deg. Σ . Alternatively, a screen may be placed in the plane Σ and a CCD camera may be used to record the 10,13 With the exception of the setups of Refs. that intensity distribution on the screen. In both cases, this could in principle be extended to DLS, these appara- geometry prevents a CCD camera to be placed directly tuses are unfit for simultaneous static and dynamic light in the plane Σ, making DLS measurements impossible. scattering at multiple angles, either because the angu- In addition, detectors and lenses used in this configu- 9,11,12 lar range is sampled sequentially, as in Refs. , or ration must often be custom-made to efficiently remove because photodiodes that cover a very large number of the transmitted beam and to limit aberrations associated 4–6 speckles are used, as in Refs. , a design not appropri- with large scattering angles. The scheme of Fig. 1b, de- 2 ate for DLS . It is worth noting that photodiode ar- scribed in Refs.5,6, is equivalent to the top one, provided rays are typically larger than CCD or CMOS detectors: that one replaces f by the sample-detector distance dΣ in this makes it difficult to transpose the optical layout of the calculation of θ16. The advantage is that large scat- Refs.4–6 to a CCD-based apparatus and would impose tering angles may be attained simply by reducing dΣ, extra constraints on the cell dimensions, in particular its without changing f and with no stringent requirements thickness L, as we shall discuss it in the following. Ad- on the numerical aperture of the Fourier lens. However, ditionally, photodiode arrays can easily accommodate a the sample thickness L must satisfy L < L1 17 a) Σ a ray-tracing software (Optical Ray Tracer 2.8 ) to test the performances of the real lenses and to fine tune the S SCREEN position of the sample and that of the third pair of lenses. OR PD To avoid vignetting, the sample must be close enough to L1a, while a minimum distance of about 5 mm is im- f posed by the thickness of the cell wall and the sample and lens holder. Using the ray-tracing software, we de- L1 termine the optimum sample-L and L -CCD distances b) 1a 3b S to be 17 mm and 30 mm, respectively. With this layout, the maximum relative spread of the spot formed on the SCREEN CCD plane by light scattered at the same θ is 1.7%, for OR PD θ = 25 deg and a 10 mm thick sample. dΣ The light source is an intensity-stabilized, linearly po- larized He-Ne laser (117A by Spectra-Physics) that op- L1 L2 c) erates at a wavelength λ = 632.8 nm, and at a power Σ of 1.0 mW. The laser beam is coupled to a polarization- S maintaining single-mode fiber optics (LPC-04-633-4/125- CCD P-0.9-4.5AS-60-A3A-3A-2 by OZ Optics), through which only the TEM00 mode can propagate. A set screw is used BS to partially block the beam before coupling it to the fiber, so that the beam power can be attenuated as required. p1 q1 The beam exiting the fiber is collimated to a diameter of 0.9 mm and it is directed on a beam splitter (BS) that FIG. 1. Scheme of common SALS and USALS setups. S: partially forwards it onto a photodiode (PDM), which sample; L1, L2: lenses; PD: photodiode array; Σ: focal plane monitors any fluctuations in the incident power for fur- of L1, BS: beam stop. ther data normalization. The transmitted component im- pinges onto the sample, which is typically contained in a parallelepiped cell of thickness 2, 5 or 10 mm (Hellma). A III. THE MALS SETUP small mirror M is placed in the focal plane Σ to intercept the transmitted beam and direct it to a second photo- A scheme of our MALS setup is shown in Fig. 2. De- diode (PDT), allowing for the measurement of the sam- ple transmission and preventing unscattered light from tails of the various optical and opto-mechanical compo- 3 nents and of their position are provided in the Supple- reaching the CCD sensor. Following Ref. , the mirror is mentary Information. In order to increase the numerical obtained by cutting at 45 deg and polishing a drill bit of aperture of the various elements, we use pairs of identi- thickness 300 µm. To minimize light reflections, all lenses cal lenses placed in contact, for which the effective focal are antireflection coated and are tilted by a few degrees length is half of that of each lens, thereby doubling the with respect to the optical axis. The same precaution is numerical aperture. The apparatus maps the cone of taken for the scattering cell. The cell is mounted on a light scattered at a given angle θ onto a ring of CCD custom-designed holder that allows one to vary both its pixels centered around the transmitted beam position. height with respect to the optical axis and its distance Functionally, it may be divided into two parts. The first from L1a. The cell can be removed from the holder and replaced exactly at the same position, i.e. for taking an part is formed by the two pairs of lenses L1a,L1b and optical background as described later. The CCD camera L2a,L2b, which are confocal and act as an (inverted) tele- scope. Their purpose is two-fold: on the one hand, they (Pulnix TM-1300) has a pixel matrix of 1285 × 1029 pix- demagnify the angle of the light scattered by the sam- els, 6.7 µm in size. A frame grabber (Meteor II Digital ple, thus mapping the MALS range to a more manage- PCI by Matrox) is used to control the camera and trans- able, lower-angle range. On the other hand, a beam stop fer the images to a PC. The whole instrument is placed can be conveniently placed in the focal plane Σ common under a box made of high density expanded polystyrene to both pairs, thereby removing the transmitted beam. panels, to minimize temperature fluctuations and to pro- tect the setup from ambient light. The second part comprises the pair of lenses L3a,L3b that act as a Fourier lens, in whose back focal plane a CCD detector is placed. The focal length of the various lenses is f = f = 62.9 mm, f = f = 150 mm 1a 1b 2a 2b IV. DATA ACQUISITION AND PROCESSING and f3a = f3b = 75.6 mm. The resulting demagnifica- tion ratio of the inverted telescope is f2a/f1a ≈ 2.4 : 1. In order to cover the MALS range, we use plano-convex We use a custom C++ code to process the CCD images lenses and work well beyond paraxial propagation; there- in real time, in order to perform both static and dynamic fore, the thin lens approximation does not apply. We use light scattering measurements. 4 L1a L1b L2a L2b L3a L3b Σ CELL BS M CCD PDM LPDT OF LASER PDT PC FIG. 2. The Mid-Angle Light Scattering apparatus described in this paper. The two pairs of lenses L1a, L1b and L2a, L2b are confocal and act as an (inverted) telescope. The beam stop M is placed in their common focal plane Σ to remove the transmitted beam. The pair of lenses L3a, L3b acts as a Fourier lens and the CCD detector is placed in its back focal plane. OF: optical fiber; BS: beam splitter; PDM, PDT: monitor and transmitted photodiode, respectively; PC: personal computer. The optical elements are drawn in scale, the cell-CCD distance is 28.8 cm. A. Static Light Scattering or imperfections on the cell walls and the lenses. To this end, the cell is filled with the solvent alone and the Data acquisition and processing for SLS involves four optical background signals for the photodiodes and the OB OB steps: i) acquisition of the electrical background, ii) ac- CCD are recorded in analogy to step i): M , T and OB quisition of the optical background, iii) acquisition of the S (q,texp). Before step iii), the cell is emptied from scattered intensity pattern, iv) applying to iii) corrections the solvent and refilled with the sample. For the optical for the contribution of i), ii) and geometrical factors in background correction to be effective, the cell should be order to obtain the scattered intensity profile I(q). Dur- kept exactly in the same position, which can be achieved in situ ing step i), the laser beam is blocked so as to measure by emptying and refilling it . When this is not pos- the contribution of electrical noise to the data. The sig- sible (e.g. for samples that need to undergo a controlled 18 nals of the monitor and transmitted photodiodes, M EB thermal history, as in Ref. ), we use a custom-designed and T EB, are averaged over 100 readings and recorded. holder that allows the cell to be removed and replaced Similarly, a series of CCD images taken at various expo- with a positioning tolerance of about 10 µm. During sure times texp are recorded, about 20 images being av- step iii), the intensity scattered by the sample at time t eraged for each texp. The images are processed in order is recorded, together with the corresponding photodiode to calculate the dark background averaged over annuli signals: S(q,texp,t), M(t) and T (t). In order to reduce of pixels centered around the transmitted beam position: noise, a few images for each texp may be averaged. De- EB EB EB pending on the number of averages and that of the ex- S (q,texp) = Sp (texp) , where Sp (texp) is the p∈Aq posure times, one full acquisition may require as little CCD electrical background signal for the p−th pixel at as 0.1 sec. Note that the use of different exposure times a given exposure time and h· · ·i indicates an aver- p∈Aq is often mandatory, given the limited dynamic range of age over the annulus of pixels associated with a given q CCD detectors and the steep variation of I(q) typically vector. Pixels that are covered by the beam block are observed over the MALS range. Data stored in steps i) software-masked and excluded from the analysis. The to iii) are processed in real time in order to calculate the purpose of step ii) is to measure any contributions to q dependent scattering intensity according to the scattered light due to flare, originating from dust 1 T (t) − T EB I(q,t)= S(q,t∗ ,t) − SEB(q,t∗ ) − SOB(q,t∗ ) − SEB(q,t∗ ) . (1) LN(q)t∗ (M(t) − M EB) exp exp T OB − T EB exp exp exp In the above equation, all optical signals are corrected mission, (T (t) − T EB)/(T OB − T EB), because the flare for the corresponding electrical background and normal- contribution is decreased when part of the incident beam ∗ ized with respect to the incident power as measured by is scattered by the sample. In Eq. (1), texp is the best the monitor photodiode. Before subtracting the opti- exposure time for a given q, allowing for a good signal cal background (the term between square brackets in the to noise ratio while preventing pixel saturation. We find ∗ r.h.s. of Eq. (1)), we correct it for the sample trans- that for a 8-bit CCD texp should be chosen as the largest 5 ∗ exposure time such that S(q,texp) . 40 grey levels. For larger values, the number of saturated pixels becomes non-negligible, while for smaller values data are affected 0.5 10 mm by digitalization noise. Note that the intensity is normal- 0.04 5 mm 2 mm ized with respect to cell thickness L and exposure time 0.4 ∗ 0.02 texp, and by the term N(q), which accounts for the av- / erage value of the solid angle associated to each pixel of 0.3 d = 14 m m 2 0.00 d = 15 m m -5 0 5 10 15 20 25 a given annulus, as well as for the dipole term sin ψp 0.2 that decreases the scattered intensity out of the scatter- sgn(x) (deg) d = 16 m m d = 17 m m 2 0.1 ing plane , where ψp is the angle between the polariza- d = 18 m m sin d = 19 m m tion of the incident light and the propagation direction 0.0 of the scattered light that reaches the p-th pixel. More d = 20 m m d = 21 m m 2 -0.1 specifically, N(q) =< ∆Ωp sin ψp) >p∈Aq , with ∆Ωp = d = 22 m m 2 2 2 d = 23 m m sin θp∆θp∆ϕp the solid angle, sin ψp =1−sin ϕp sin θp -0.2 the dipole factor, θp the scattering (polar) angle, and ϕp -2 0 2 4 6 8 the azimuthal angle. For a pixel p with coordinates (x, y) sgn(x)r (mm) with respect to the transmitted beam position one has 1 dθp ∆θp = (x∆x + y∆y) (2a) r dr FIG. 3. MALS angular calibration performed using a diffrac- dϕp dϕp tion ruling with m = 1000 lines cm−1. In the main plot the ∆ϕp = ∆x + ∆y , (2b) dx dy sine of the angular position of the the maxima of the diffracted beam is shown as a function of the maxima position on the 2 2 with r = x + y , ϕp = arctan(y/x), and ∆x and ∆y CCD plane, for distances d between the diffraction grating the pixel dimensions.p The functional form relating θp to and lens L1a between 14 and 23 mm. The symbols are the dθ data, the lines are fits according to Eq. (5). The inset shows r and from which the term p can be calculated will be dr the loss of angular resolution, as defined in the text, due to a discussed in Sec. V. sample thickness of 2 mm,5 mm and 10 mm. B. Dynamic Light Scattering where feff = f3af2a/(2f1a) ≈ 15.9 mm is the effective focal length of the Fourier lens, which accounts for both The same four steps discussed in the previous subsec- the use of pairs of lenses and the demagnification of the tion are also required for DLS measurements, although inverted telescope. Note that for the moment we have of course the data processing in step iv) is different. We neglected refraction, assuming that the scatterers are in use a software multi-tau correlator19, whose details are air. Since deviations are to be expected with respect to discussed in Ref.8, together with the algorithm for cor- this simple formula, we perform an angular calibration recting for the electrical and optical background. Here, of the setup using a diffraction ruling with m = 1000 we simply recall that the correlator calculates in real lines cm−1. The diffraction grating is positioned perpen- time the time autocorrelation function of the CCD signal, dicular to the incident beam, so that the maxima of the diffracted beam occur at angles such that sin θk = mkλ, hSp(t)Sp(t + τ)ip∈A . After correction for the electri- q t k = 0, ±1, ±2.... We vary the distance d between the calD and optical backgroundsE and proper normalization, 2 diffraction grating and L1a between 14 and 23 mm in using the Siegert relation one obtains the field autocor- steps of 1 mm and for each d we measure the position relation function r of the maxima on the CCD sensor. The results are hE(q,t)E∗(q,t + τ)i shown in Fig. 3. The highest order observable is k = 7, g (q, τ)= t , (3) 1 hI(q,t)i corresponding to θk = 26.3 deg, which we achieve by off- t setting the CCD with respect to the optical axis, such which is directly related to the sample dynamics2. that the transmitted beam impinges near one corner of the detector, rather than in its center. The distance d for which the maxima appear to be sharper is found V. ANGULAR CALIBRATION to be 18 mm, close to d = 17 mm as evaluated using the ray-tracing software. Thus, thick samples are always With the optical layout of Fig. 2 and in the thin lens positioned such that the cell center is at 18 mm from and paraxial approximations, the relationship between the first lens. Figure 3 shows that sin θk varies approx- the scattering angle θ and the distance r from the optical imately linearly with r, as expected from Eq. (4) given 2 2 axis of a CCD pixel is that feff >> r . However, deviations are observed at large r; more importantly, these deviations depend on 2 2 −0.5 sin θ = r(r + feff) , (4) d, implying that lens aberrations limit the angular res- 6 olution of the setup for samples of finite thickness. To quantify these effects, we fit for each d the data of Fig. 3 1 using the empirical relation 10 -3 10 0 1 1 10 sin θk = ar + − , (5) (arb.un.) I b − r b d = 14 m m -1 10 -4 d = 15 m m 10 3 4 5 d = 16 m m with a and b d-dependent fitting parameters and where -1 -2 q (µm ) d = 17 m m r is measured in mm. As seen in Fig. 3, this expression 10 d = 18 m m is in excellent agreement with the data. For samples of d = 19 m m -3 (arb.un.) 10 I d = 20 m m finite thickness and for a solvent of refractive index n, we d = 21 m m define the scattering angle associated to a pixel p laying -4 d = 22 m m 10 at a distance r from the optical axis as d = 23 m m theory -5 10 1 1 1 0.1 1 θp = arcsin ar¯ + − , (6) n ¯b − r ¯b -1 q (µm ) where the n−1 factor accounts for refraction at the solvent-wall and wall-air interfaces and wherea ¯ and ¯b are the average of the fitting parameters a and b over FIG. 4. Diffraction pattern of a pinhole with a nominal di- the range of d corresponding to the sample thickness. ameter of 10 µm measured for ten values of the sample-L1a −1 For a 10 mm-thick cell, we finda ¯ = 17.27 mm, about distance d. In the inset, the discrepancy in the q-calibration 8.6% larger than feff as calculated neglecting aberrations. and the intensity level for the ten distances at high q vectors The q vector associated with an annular set of pixels is are visible. An image of the pinhole taken with an optical obtained by averaging the corresponding value for each microscope is also shown. pixel of the annulus: 4πn θ q = sin p (7) covered by the instrument, eighteen exposure times have λ 2 p∈Aq been used, from 0.08 ms to 80 ms, with a scaling factor of 1.5 between one exposure time and the next one. with θp as obtained from Eq. (6). To verify the q calibration and to check for the angu- The inset of Fig. 3 quantifies the loss of angular res- lar response of the apparatus, we measure the diffraction olution due to sample thickness by showing the relative pattern of a pinhole with a nominal diameter of 10 µm. standard deviation of θ, σθ/θ, associated with the stan- Inspection under an optical microscope reveals a real di- dard deviation of the coefficients a and b over the range of ameter of (8.66 ± 0.24 µm) and some deviations from a d corresponding to representative values of the cell thick- perfectly circular shape (inset of Fig. 4). The diffrac- ness L. For L ≤ 5 mm, the relative angular spread is well tion pattern of the pinhole was measured by placing it below 1% over the full range covered by the setup, while at ten distances between 14 mm and 23 mm from L1a for a 10 mm-thick cell, the loss of angular resolution is and centering it onto the optical axis. The beam stop about 4% at the largest θ. was removed. The ten intensity distributions are shown in Fig. 4 (symbols, data rescaled so that I(q → 0) = 1), together with the Airy function predicted by the Fraun- VI. TESTS OF THE APPARATUS PERFORMANCES hofer diffraction theory (line). All curves agree well with the theory up to q ≈ 2.5 µm−1. Beyond this value, the We perform a series of experiments to validate the an- q calibration is still good up to q = 5 µm−1, but de- gular calibration determined in Sec. V, determine the viations appear with respect to the theory. While this angular response of the apparatus and to test its per- discrepancy may be due in part to the shape defects of formances for DLS. the pinhole, we observe that the intensity level is system- atically lower than the theoretical one for d ≤ 16 mm and systematically higher for d ≥ 20 mm, as better seen in A. Static light scattering the inset of Fig. 4. To better quantify the d-dependent response of the Static light scattering tests were performed on repre- setup, we measure the scattering from a ground glass, sentative 2-dimensional and 3-dimensional samples. For a 2-dimensional scatterer, as a function of the sample- all samples, I(q) was measured for pixel annuli whose L1a distance, for 14 mm ≤ d ≤ 23 mm, in steps average radius ranged from 29 pixels to 1279 pixels, for of 1 mm. The results are shown in Fig. 5. Up to a total of 126 values equally spaced in a linear scale. q = 1 µm−1 no significant differences are observed. For The thickness of the annuli is 1 pixel. To cope with the 1 µm−1 ≤ q ≤ 4.6 µm−1, the intensity level is systemat- large variation of the scattered intensity over the range ically lower for d = 14 mm and d = 15 mm, as compared 7 0 d = 14 mm 10 2 d = 15 mm 10 d = 16 mm d = 17 mm d = 18 mm 1 -1 10 1. 4 d = 19 mm 10 1. 2 d = 20 mm 1. 0 d = 21 mm 0 0. 8 d = 22 mm 10 (arb.un.) 0. 6 (arb.un.) d = 23 mm -2 I I d=18 10 0. 4 I/I PMMA, thi ckness: 5 m m 0. 2 -1 PMMA, thi ckness: 2 m m 0. 0 10 1 2 3 4 5 6 Pol ydi sperse Mie - 1 q (µm ) -3 10 0.1 1 0.1 1 -1 -1 q (µm ) q (µm ) FIG. 5. Scattering from a ground glass positioned at distances FIG. 6. Intensity distribution measured for a sample of d varying between 14 mm and 23 mm. The inset shows the PMMA spheres with an average diameter of 2.09 µm sus- ratio of the measured intensity with respect to that recorded pended in a mixture of decalin and tetralin and placed in two for d = 18 mm, corresponding to the optimum sample-L1a cells of thickness 5 mm and 2 mm. Both datasets have been distance. rescaled so that I(q → 0) = 1. The black line is the form factor calculated using the Mie theory. to the other curves, while it is systematically higher for d ≥ 21 mm, confirming the trend observed for the pin- and the PMMA refractive index is nPMMA ≈ 1.501, as hole. For d = 23 mm an anomalous increment in the estimated by preparing samples in slightly different de- −1 intensity profile is visible at q =4.6 µm and the same calin/tetralin mixtures and by taking nPMMA as the re- happens at higher q vector for smaller distances. This fractive index of the solvent for which the scattered in- extra-signal is due to aberrations affecting rays scattered tensity is minimal. The suspension was filtered through at high q vectors and impinging close to the lens edge. a 5 µm PTFE membrane and then placed in two cells The ray-tracing software shows that these rays are sys- of thickness 5 mm and 2 mm, respectively. In the first tematically bent towards the optical axis, resulting in a case, the sample volume fraction was ϕ =0.2% while for spurious increase of the measured I(q). At even larger q, the thinner cell it was ϕ =0.4%. Fig. 6 shows the inten- the intensity decreases steeply because of vignetting. To sity distributions measured for the two samples. Data quantify the discrepancy between the ten intensity distri- have been rescaled so that I(q → 0) = 1. The line is butions, the ratio between each curve and the intensity the form factor from the Mie theory1 applied to polydis- profile measured for d = 18 mm is shown in the inset of perse spheres, calculated using the MiePlot software20, Fig. 5. If we consider the five distances between 16 mm for the size distribution obtained by TEM. The agree- and 20 mm, the ratio varies between 0.87 and 1.07 (hor- ment between the data and the theory is very good up to izontal black lines in the inset), and the maximum ac- 4 µm−1. At higher q, the intensity distribution is system- cessible scattering vector is 5 µm−1. Thus, the response atically larger than the theoretical curve. This discrep- of the instrument for a 5 mm thick sample centred in ancy is unlikely to stem mainly from a non-uniform angu- d = 18 mm is flat to within ≈ ±10%. If we consider lar response of the setup, since the tests with the pinhole the two distances 18 mm and 19 mm (corresponding to a and the ground glass suggest that deviations should com- 2 mm-thick sample) the ratio deviates from 1 by less than pensate once integrated over the thickness of the sam- 5% and the maximum accessible q vector is 5.2 µm−1. ple. Other possible explanations include deviations from We test the setup performances on a typical 3- a perfectly spherical particle shape (as observed in the dimensional sample by measuring the scattering from TEM images), scattering from the PHHS corona that a suspension of poly(methyl methacrylate)(PMMA) may become non-negligible when matching closely the spheres with an average diameter of 2.09 µm and relative PMMA refractive index, and multiple scattering. Con- polydispersity of 2.5%, as obtained from transmission cerning the latter point, we note that although the sus- electron microscopy (TEM). The particles are sterically pension transmission was very high (90%), even modest stabilized by a thin layer of chemically grafted poly- 12- amounts of multiple scattering would tend to increase hydroxystearic acid (PHSA), and are suspended in a mix- I(q) in those angular ranges where the signal is markedly ture of decalin and tetralin (0.676/0.324 v/v at 20◦C). lower than for nearby q vectors, as observed here. The measurements have been performed at 26.5◦C; at To further test the angular response of the instrument this temperature the solvent refractive index is 1.492 for q ≥ 4 µm−1, we measure I(q) for fractal aggregates 8 -1 q = 0.29 µm -1 1.0 q = 0.38 µm 1 -1 10 q = 0.49 µm -1 q = 0.64 µm -1 0.8 q = 0.83 µm -1 q = 1.08 µm 1. 0 -1 q = 1.41 µm 0 -1 10 q = 1.84 µm 0.6 0. 8 cel l thickness: 5 m m -1 q = 2.41 µm 1 (arb.un.) 0. 6 -1 1 cel l thickness: 2 m m g I q = 3.18 µm g -1 q = 4.22 µm 0.4 power l aw fit 0. 4 -1 q = 5.71 µm -1 0. 2 10 exponential 2 3 4 5 6 0.2 0. 0 0 1 2 3 10 10 10 10 -1 q (µm ) t (s) 0.0 -1 0 1 2 10 10 10 10 2 -2 tq (s µm ) FIG. 7. Intensity distributions measured for fractal aggre- gates of latex spheres placed in cells of thickness 5 mm and 2 2 mm. The plot zooms into the high q range 1 µm−1 = 10 −1 q = 6 µm . The black curve is a power law with exponent 1 10 −df = −2.3, obtained by fitting the two experimental curves − − 1 1 0 in the range 1.5 µm ≤ q ≤ 3.5 µm 10 (s) MALS t -1 goniometer 10 fit -2 10 made of colloidal particles. Fractal aggregates provide a -1 0 1 convenient means to test the apparatus, since the scatter- 10 10 10 -1 ing profile has a simple power-law dependence on scatter- q (µm ) 14 −d ing vector : I(q) ∼ q f , with df the fractal dimension of the aggregate. While the fractal regime only extends beyond the particle size and below the aggregate dimen- − sion, it is easy to have the I ∼ q df regime covering the FIG. 8. a) The twelve correlation functions for a sample of whole MALS range, by choosing appropriately the par- polystyrene spheres with diameters of 530 nm diluted in a mix ticle size and the aggregation time. To form the aggre- of water/glycerol 0.3w/0.7w are shown as a function of time rescaled by q2. All data collapse onto a single master curve gates, we start by preparing a particle suspension of latex that is well fitted by an exponential decay correlation (black spheres with a diameter of 19 nm in a buoyancy match- line). The inset shows the twelve original correlation func- ing mixture of Milli-Q water and D2O (0.5/0.5 v/v), at tions on a semilogarithmic plot. b) The decay time obtained −4 ϕ =6 × 10 . Separately, a 30 mM solution of MgCl2 is by fitting each correlation function to an exponential decay, as prepared, using the same solvent. Equal amounts of the a function of q, is shown in a double logarithmic scale (stars). particle suspension and the salt solution are mixed di- The circles are the decay time measured for the same sam- rectly in the scattering cell (cells with L = 5 mm and ple but at larger scattering angles using a conventional DLS L = 2 mm were used), just before starting measure- apparatus. ments that are run for about 17 hours. Initially, I(q) exhibits a power law tail at large q and a rolloff at lower q (data not shown), corresponding to the size of the ag- To summarize, the SLS tests indicate that the angular gregates. As clusters keep growing, the rolloff moves to response of the setup is flat for thin samples (L ≤ 2 mm) smaller q, eventually exiting from the MALS window. placed at d = 18 mm, while for thicker samples I(q) −1 Figure 7 shows the intensity distribution measured in the tends to be overestimated beyond q ≈ 4 µm . Even in two scattering cells at later stages, focussing on the range the worst case that we have tested (a 5-mm thick cells), 1 µm−1 ≤ q ≤ 6 µm−1. The black curve is a power law deviations do not exceed 24% for q vectors as large as −1 with exponent −df , obtained by fitting the two exper- 5 µm . imental curves in the range 1.5 µm ≤ q ≤ 3.5 µm−1, where the previous tests indicate that the instrument re- B. Dynamic Light Scattering sponse is flat. We find df = 2.3, consistent with typical values for colloidal aggregates14. The fit is in excellent agreement with the data below q ≤ 3.5 µm−1. The agree- To test the DLS capabilities of our apparatus, we mea- ment remains quite good up to q ≈ 5 µm−1, where a sure the field time autocorrelation function of submi- higher than expected intensity is measured, due to aber- cron Brownian particles, for which one expects g1(τ) = rations. At q =5 µm−1 deviations are 15% and 24% for exp −Dq2τ 2, where D is the particle diffusion coeffi- the L = 2 mm and L = 5 mm cells, respectively. cient. Twelve annuli of pixels were processed simultane- 9 ously in real time, at a rate of 4 images per second and effect to be even more relevant right after filling the cell, using a single exposure time texp = 5 msec. The annu- when temperature fluctuations and gradients are not sta- lus average radius ranged from 70 pixels to 1253 pixels, bilized yet. Such ballistic motion, while negligible with logarithmically spaced with a scaling factor of 1.3 be- respect to diffusion on the small length scales probed by tween one annulus and the next one. The width of the traditional DLS, become increasingly important as q is innermost annulus was 1 pixel, corresponding to a rela- decreased, since for ballistic motion the relaxation time −1 −2 tive spread in scattering vector δq/q = 1.4%. The same of g1 scales as q , as opposed to q for Brownian dif- ratio was kept for all other annuli. With this annuli se- fusion2. Therefore, their impact becomes eventually visi- lection, the time correlation function is measured in the ble, as observed here in the low q spectrum of the MALS range 0.29 µm−1 9P. A. Webb, A primer on particle sizing by static laser light 16J. W. Goodman, Introduction To Fourier Optics (Roberts and scattering, Micromeritics Instrument Corp. 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Multiangle Static and Dynamic Light Scattering in the Intermediate
