Static Light-scattering Measurements Research Articles

Imino-bis-propane diol functional polymer for efficient boron removal from aqueous solutions via continuous PEUF process

Boron contamination is an environmentally serious problem especially due to its hazards on plants. In this work, newly synthesized poly (vinyl amino-N, N′-bis-propane diol) (GPVA) is presented as an excellent chelating polymer for boron removal. Boron concentration could be reduced from 10ppm down to 0.4ppm via total recycle Polymer Enhanced Ultrafiltration (PEUF). Experiments revealed that the polymer shows the highest PEUF boron rejections (96% at pH 9.0) reported so far, without precipitation in a wide concentration range. Detailed investigations such as effect of pH, loading (boron-to-polymer mass ratio) and polymer concentration showed that, the boron binding is highly pH dependent and increases with boron concentration at constant loading. Furthermore, permeate flux was found as nearly independent from polymer concentration up to 10g/L at 200kPa pressure drop. In addition, gyration radius of the polymer in solution increases while increasing pH and polymer concentration in the presence of boron as inferred from Dynamic and Static Light Scattering measurements. Regarding with high efficiency and hydrolytic stability of GPVA, its use in combination with PEUF would be of interest for large scale and long term use in boron removal.

Structural properties of thermoresponsive poly(N-isopropylacrylamide)-poly(ethyleneglycol) microgels

We present investigations of the structural properties of thermoresponsive poly(N-isopropylacrylamide) (PNiPAM) microgels dispersed in an aqueous solvent. In this particular work poly(ethyleneglycol) (PEG) units flanked with acrylate groups are employed as cross-linkers, providing an architecture designed to resist protein fouling. Dynamic light scattering (DLS), static light scattering (SLS), and small angle neutron scattering (SANS) are employed to study the microgels as a function of temperature over the range 10 °C ≤ T ≤ 40 °C. DLS and SLS measurements are simultaneously performed and, respectively, allow determination of the particle hydrodynamic radius, R(h), and radius of gyration, R(g), at each temperature. The thermal variation of these magnitudes reveals the microgel deswelling at the PNiPAM lower critical solution temperature (LCST). However, the hydrodynamic radius displays a second transition to larger radii at temperatures T ≤ 20 °C. This feature is atypical in standard PNiPAM microgels and suggests a structural reconfiguration within the polymer network at those temperatures. To better understand this behavior we perform neutron scattering measurements at different temperatures. In striking contrast to the scattering profile of soft sphere microgels, the SANS profiles for T ≤ LCST of our PNiPAM-PEG suspensions indicate that the particles exhibit structural properties characteristic of star polymer configurations. The star polymer radius of gyration and correlation length gradually decrease with increasing temperature despite maintenance of the star polymer configuration. At temperatures above the LCST, the scattered SANS intensity is typical of soft sphere systems.

Facile synthesis of nanoparticles via assembly of photopolymerized and self-crosslinked random copolymers in selective organic media

We report the facile synthesis of vesicular nanoparticles via self-assembly of random copolymers in selective organic solvents. The polymers were synthesized via photopolymerization in bulk from acryloyl chloride (AC) containing a small amount of hydrolyzed acrylic acid (AA) at ambient condition. Fourier transform infrared spectroscopy of the photopolymerized product revealed two main chemical components: poly(acryloyl chloride) (PAC) and acid anhydride. The later peak intensified when increasing the initial AA composition in the monomer solution, suggesting that the hydrolyzed AC contributed to the formation of crosslinked anhydride, leading to amphiphilicity of the polymer with solvophobic anhydride and solvophilic PAC chains. At an optimal UV dosage (2000 mJ cm−2), vesicular nanoparticles (∼90 nm in diameter) were obtained from polymers assembled in acetone. The particle size and morphology were confirmed by scanning electron microscopy, transmission electron microscopy, and dynamic and static light-scattering measurements. Further, we found that the random copolymers self-organized into vesicles in merely good solvents of PAC chains, including acetonitrile, tetrahydrofuran, and 1,4-dioxane, but became aggregated and precipitated out in poor solvents of PAC, such as isopropanol, ethanol, toluene, xylene, and hexane. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012

Determination of the Molar-Mass Averages of Random Poly(aspartate-co-lactide) Copolymers by Tuning the Ionic Strength of the Solvent

Water-soluble sodium poly(aspartate-co-lactide) (PALNa) copolymers with a molar ratio of aspartate-to-lactide units equal to 1:0.6, 1:1.0 and 1:1.5 were studied using NMR spectroscopy to determine the composition as well as SEC-MALS and static light-scattering measurements to determine the molar-mass characteristics of the copolymers. In the copolymer aqueous solutions, high-molar-mass species were detected, most probably due to the incomplete dissolution of the samples. The molar-mass averages determined in water with added simple electrolyte, i.e., NaCl, were much lower than the values determined in pure water. The concentration of the salt, which allows dissolution on a molecular level, and the separation predominantly according to a size-exclusion mechanism depend on the chemical composition of the PALNa copolymers. The optimal mobile phase for the PALNa-1/0.6 and the PALNa-1/1.0 copolymers was 0.1 M NaCl at pH 9, and for the PALNa-1/1.5 copolymer with a higher content of lactide units it was 0.05 M NaCl at pH 9. The molar-mass averages of the PALNa-1/1.0 copolymer, determined by SEC-MALS and static light-scattering measurements, were comparable.

Amino-ethoxilated fluorinated amphiphile: Synthesis, self-assembling properties and interactions with ssDNA

Fluorinated compounds are not only more hydrophobic than their hydrogenated counterparts but also oleophobic. This makes them interesting for biomedical applications. They form a large variety of supramolecular architectures and, if appropriately designed, they spontaneously form vesicles. Herein we report the synthesis of novel monocatenar fluorinated surfactant, named F9NH(EO)2NH2, its use as ssDNA vector with potential applications in gene delivery or RNA interference. The molecular structure of the amphiphilic compound was designed in three parts: (i) a fluorinated hydrophobic tag responsible of self-assembling, (ii) a flexible and hydrophilic ethylenoxide moiety as polar head and (iii) a protonated at physiological pH primary amine responsible of binding. Surface tension, dynamic light scattering (DLS) measurements, and TEM studies showed that the monocatenar surfactant F9NH(EO)2NH2 spontaneously self-assembles into vesicles. This represents another example in addition to the few existing in the literature and it highlights the unique properties of the fluorinated tail. Interactions between ssDNA and cationic monocharged F9NH(EO)2NH2 were probed by static light scattering (SLS), fluorescence and circular dichroism (CD). F9NH(EO)2NH2 was found to exclude ethidium bromide (EB) from ssDNA/EB complex, and this process was dependent on the ssDNA concentration and on the charge ratio surfactant to ssDNA. CD spectra showed that by increasing the surfactant concentration, ssDNA undergoes a continuous conformational transition from native B-form to a more ordered phase (Z+/Z−≤3) and then to a less ordered phase (Z+/Z−≥4), which confirmed the results obtained by fluorimetry and SLS measurements. Cell viability studies reveled that monocatenar fluorinated surfactant F9NH(EO)2NH2 is not toxic for human cells (MRC-5) below 40μM.

Crystal Structure and Role of Glycans and Dimerization in Folding of Neuronal Leucine-Rich Repeat Protein AMIGO-1

AMIGO-1 is the parent member of a novel family of three cell surface leucine-rich repeat (LRR) proteins. Its expression is induced by the binding of HMGB1 (high-mobility group box 1 protein) to RAGE (receptor for advanced glycation end products) on neurons. Binding of HMGB1 to RAGE is known to have a direct effect on cellular growth regulation and mobility, and AMIGO-1 directly supports growth of neuronal processes and fasciculation of neurites. In addition, the second member of the AMIGO-family, AMIGO-2, has been implicated in adhesion of tumor cells in adenocarcinoma and survival of neurons.We have determined the crystal structure of AMIGO-1 at 2.0 Å resolution, which reveals a typical cell surface LRR domain arrangement with N- and C-terminal capping domains with disulfide bridges, followed by a C2-type Ig domain. AMIGO-1 is a dimer, with the LRR regions forming the dimer interface, and sequence conservation analysis and static light-scattering measurements suggest that all three AMIGO family proteins form similar dimers. Based on the AMIGO-1 structure, we have also modeled AMIGO-2 and present small-angle X-ray scattering data on AMIGO-2 and AMIGO-3. Our mutagenesis studies show that AMIGO-1 dimerization is necessary for proper cell surface expression and thus probably for proper or stable folding in the endoplastic reticulum and for the function of the protein. Based on the data presented earlier, we also suggest that dimerization through the LRR–LRR interface is likely to be involved in cell–cell adhesion by AMIGO-1, while extensive glycosylation may have a role.

High-Throughput Analysis of Concentration-Dependent Antibody Self-Association

Monoclonal antibodies are typically monomeric and nonviscous at low concentrations, yet they display highly variable associative and viscous behavior at elevated concentrations. Although measurements of antibody self-association are critical for understanding this complex behavior, traditional biophysical methods are not capable of characterizing such concentration-dependent self-association in a high-throughput manner. Here we describe a nanoparticle-based method, termed self-interaction nanoparticle spectroscopy, that is capable of rapidly measuring concentration-dependent self-interactions for three human monoclonal antibodies with unique solution behaviors. We demonstrate that gold nanoparticles conjugated with antibodies at low protein concentrations (<40 μg/mL) display self-association behavior (as measured by the interparticle distance-dependent plasmon wavelength) that is well correlated with static light-scattering measurements obtained at three orders of magnitude higher antibody concentrations. Using this methodology, we find that the antibodies display a complex pH-dependent self-association behavior that is strongly influenced by the solution ionic strength. Importantly, we find that a polyclonal human antibody is nonassociative for all solution conditions evaluated in this work, suggesting that antibody self-association is more specific than previously realized. We expect that our findings will guide rational manipulation of antibody phase behavior, and enable studies that elucidate sequence and structural determinants of antibody self-association.

Use of dynamic light scattering to determine second virial coefficient in a semidilute concentration regime

The present work discusses an alternative procedure to obtain static light scattering (SLS) parameters in a dilute and semidilute concentration regime from a dynamic light scattering (DLS) instrument that uses an avalanche photodiode (APD) for recording the scattered intensity signal. An APD enables one to perform both SLS and DLS measurements by photon counting and photon correlation, respectively. However, due to the associated recovery time, the APDs are susceptible to saturation (above 1000kcps), which may limit the measurements in systems that scatter too much light. We propose an alternative way of obtaining the SLS parameters with instruments that use APD for recording signal intensities.

Integrated biophysical studies implicate partial unfolding of NBD1 of CFTR in the molecular pathogenesis of F508del cystic fibrosis

The lethal genetic disease cystic fibrosis is caused predominantly by in-frame deletion of phenylalanine 508 in the cystic fibrosis transmembrane conductance regulator (CFTR). F508 is located in the first nucleotide-binding domain (NBD1) of CFTR, which functions as an ATP-gated chloride channel on the cell surface. The F508del mutation blocks CFTR export to the surface due to aberrant retention in the endoplasmic reticulum. While it was assumed that F508del interferes with NBD1 folding, biophysical studies of purified NBD1 have given conflicting results concerning the mutation's influence on domain folding and stability. We have conducted isothermal (this paper) and thermal (accompanying paper) denaturation studies of human NBD1 using a variety of biophysical techniques, including simultaneous circular dichroism, intrinsic fluorescence, and static light-scattering measurements. These studies show that, in the absence of ATP, NBD1 unfolds via two sequential conformational transitions. The first, which is strongly influenced by F508del, involves partial unfolding and leads to aggregation accompanied by an increase in tryptophan fluorescence. The second, which is not significantly influenced by F508del, involves full unfolding of NBD1. Mg-ATP binding delays the first transition, thereby offsetting the effect of F508del on domain stability. Evidence suggests that the initial partial unfolding transition is partially responsible for the poor in vitro solubility of human NBD1. Second-site mutations that increase the solubility of isolated F508del-NBD1 in vitro and suppress the trafficking defect of intact F508del-CFTR in vivo also stabilize the protein against this transition, supporting the hypothesize that it is responsible for the pathological trafficking of F508del-CFTR.

Highly Enhanced Cytotoxicity of a Dimeric Bispecific Diabody, the hEx3 Tetrabody

We previously reported the utility for cancer immunotherapy of a humanized bispecific diabody (hEx3) that targets epidermal growth factor receptor and CD3. Here, we used dynamic and static light scattering measurements to show that the multimer fraction observed in hEx3 in solution is a monodisperse tetramer. The multimerization into tetramers increased the inhibition of cancer cell growth by the hEx3 diabody. Furthermore, 1:2 stoichiometric binding for both antigens was observed in a thermodynamic analysis, indicating that the tetramer has bivalent binding activity for each target, and the structure may be in a circular configuration, as is the case for the single-chain Fv tetrabody. In addition to enhanced cytotoxicity, the functional affinity and stability of the hEx3 tetrabody were superior to those of the hEx3 diabody. The increase in molecular weight is also expected to improve the pharmacokinetics of the bispecific diabody, making the hEx3 tetrabody attractive as a therapeutic antibody fragment for cancer immunotherapy.

Purification, crystallization and preliminary crystallographic analysis of very-long-chain acyl-CoA dehydrogenase from Caenorhabditis elegans.

Acyl-CoA dehydrogenase [acyl-CoA:(acceptor) 2,3-oxidoreductase; EC 1.3.99.3] catalyzes the first reaction step in mitochondrial fatty-acid beta-oxidation. Here, the very-long-chain acyl-CoA dehydrogenase from Caenorhabditis elegans (cVLCAD) has been cloned and overexpressed in Escherichia coli strain BL21 (DE3). Interestingly, unlike other very-long-chain acyl-CoA dehydrogenases, cVLCAD was found to form a tetramer by size-exclusion chromatography coupled with in-line static light-scattering, refractive-index and ultraviolet measurements. Purified cVLCAD (12 mg ml(-1)) was successfully crystallized by the hanging-drop vapour-diffusion method under conditions containing 100 mM Tris-HCl pH 8.0, 150 mM sodium chloride, 200 mM magnesium formate and 13% PEG 3350. The crystal has a tetragonal form and a complete diffraction data set was collected and processed to 1.8 A resolution. The crystal belonged to space group C2, with unit-cell parameters a = 138.6, b = 116.7, c = 115.3 A, alpha = gamma = 90.0, beta = 124.0 degrees . A self-rotation function indicated the existence of one noncrystallographic twofold axis. A preliminary molecular-replacement solution further confirmed the presence of two molecules in one asymmetric unit, which yields a Matthews coefficient V(M) of 2.76 A(3) Da(-1) and a solvent content of 55%.

Evaluation of light scattering models to characterize concentrated polymer particles embedded in a solid polymer matrix

This work deals with the application of the static light scattering (SLS) model of Vrij (VM) for the characterization of a spherical polydisperse concentrated polymer particle system. This model is the exact solution for the SLS of such mixture of particles in the Percus–Yevick approximation. The analyzed polymer particle samples are obtained by solution polymerization of isobornyl methacrylate in polyisobutylene. At the end of the polymerization, as a result of phase separation, a particle system of micrometer sized particles with a moderate distribution of sizes and a volume fraction between 5 and 10% is formed. The SLS data were also analyzed using the local monodisperse approximation (LMA), a well-known approximation to the model of Vrij. As expected, the estimations with the VM gave better results than those performed with the LMA model for the parameters related to the shape of the particle size distribution as compared with independent determinations of these quantities obtained from scanning electron microscopy micrographs. However, the main motivation to use the more rigorous model seems to be the fact that the volume fraction of particles can be extracted from the data even when relative SLS measurements are used. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 958–963, 2010

Unstable Supramolecular Structure of [Bmim][BF4] in Aqueous Solution

To unravel the exact composition and structure of aggregates in an aqueous solution of 1-butyl-3-methylimidazolium tetrafluoroborate ([Bmim][BF4 ]), we performed static and dynamic light-scattering measurements, as well as transmission electron microscopy (TEM). Results from this work show that the aggregates are vesicles and unstable; herein, we discuss the driving force behind the self-assembly. Apart from the van der Waals forces and repulsive electrostatic interactions between adjacent cation clusters, the hydrogen-bonding forces as well as counterion effects might also contribute to this driving force. The information obtained here is useful for a better understanding of the vital role that aggregation behavior plays in the field of ionic liquid recovery, and its potential use in controlled release, drug delivery, and petroleum recovery.

Phase behavior of the Pluronic P103/water system in the dilute and semi-dilute regimes

The detailed temperature–composition phase diagram of the P103/water system in the dilute and semidilute regions is reported here using density and ultrasound velocity measurements, differential scanning calorimetry (DSC), rheometry and dynamic (DLS) and static light scattering (SLS). These techniques allow a precise determination of the critical micellar temperature (CMT), the sphere-to-rod micellar transition temperature (GMT) and the cloud point temperature (CPT) as a function of concentration. DLS and SLS measurements were employed to gain information on unimers and aggregate sizes and on the transition from spherical-to-rod micelles.

Tendency toward crossover of the effective susceptibility exponent from its doubled Ising value to its doubled mean-field value near a double critical point

The critical behavior of osmotic susceptibility in an aqueous electrolyte mixture 1-propanol (1P) + water (W) + potassium chloride is reported. This mixture exhibits re-entrant phase transitions and has a nearly parabolic critical line with its apex representing a double critical point (DCP). The behavior of the susceptibility exponent is deduced from static light-scattering measurements, on approaching the lower critical solution temperatures (T(L)'s) along different experimental paths (by varying t) in the one-phase region. The light-scattering data analysis substantiates the existence of a nonmonotonic crossover behavior of the susceptibility exponent in this mixture. For the T(L) far away from the DCP, the effective susceptibility exponent gamma(eff) as a function of t displays a nonmonotonic crossover from its single limit three-dimensional (3D)-Ising value (approximately 1.24) toward its mean-field value with increase in t. While for that closest to the DCP, gamma(eff) displays a sharp, nonmonotonic crossover from its nearly doubled 3D-Ising value toward its nearly doubled mean-field value with increase in t. The renormalized Ising regime extends over a relatively larger t range for the T(L) closest to the DCP, and a trend toward shrinkage in the renormalized Ising regime is observed as T(L) shifts away from the DCP. Nevertheless, the crossover to the mean-field limit extends well beyond t>10(-2) for the T(L)'s studied. The observed crossover behavior is attributed to the presence of strong ion-induced clustering in this mixture, as revealed by various structure probing techniques. As far as the critical behavior in complex or associating mixtures with special critical points (like the DCP) is concerned, our results indicate that the influence of the DCP on the critical behavior must be taken into account not only on the renormalization of the critical exponent but also on the range of the Ising regime, which can shrink with decrease in the influence of the DCP and with the extent of structuring in the system. The utility of the field variable t(UL) in analyzing re-entrant phase transitions is demonstrated. The effective susceptibility exponent as a function of t(UL) displays a nonmonotonic crossover from its asymptotic 3D-Ising value toward a value slightly lower than its nonasymptotic mean-field value of 1. This behavior in the nonasymptotic, high t(UL) region is interpreted in terms of the possibility of a nonmonotonic crossover to the mean-field value from lower values, as foreseen earlier in micellar systems.

Static light scattering and transmission microscopy study of dilute Nafion solutions

AbstractNafion, an ionomer composed of perfluorocarbon backbone and vinyl ether side chains terminated with -SO3 - groups, has dual solubility parameters, i.e. δ1= 9.7 (cal/cm3)1/2 relating to perfluorocarbon backbones and δ2 = 17.3 (cal/cm3)1/2 relating to sulfonated vinyl ether side chains. It had been reported that Nafion molecules aggregated in dilute and semi-dilute water and alcohol solutions. In this study, we report static light scattering (SLS) measurements of dilute Nafion in N,N’- dimethyl formamide (DMF, solubility parameter δ =12.2 (cal/cm3)1/2, dielectric constant ε=36.7) and N,N’-dimethyl acetamide (DMAc, δ =10.8 (cal/cm3)1/2, ε=37.8) solutions with Nafion concentrations ranging from 0.2 mg/ml to 1.0 mg/ml. The DMF and DMAc solvents have δ closing to δ1 of Nafion perfluorocarbon backbones and ε much lower than water (ε =78). Thus DMF and DMAc are compatible with Nafion perfluorocarbon and few Nafion molecules aggregate in DMF and DMAc solvents when [Nafion]&lt; 1.0 mg/ml. The lower ε of DMF and DMAc led to low polyelectrolyte effect of Nafion in these two solvents. The obtained Mws of present work were in good agreement with the Mw obtained from Nafion in dilute dimethyl sulfoxide (DMSO, δ =12.0 (cal/cm3)1/2, ε=46.6) solutions, reported by Lousenberg, using size exclusion chromatography incorporating static light scattering detection. The &lt;RG&gt; data obtained from SLS measurements were compared with the particles sizes of transmission electron microscope (TEM) observations of freeze dried Nafion thin films prepared on copper grids with 0.6 mg/ml Nafion solutions. Comparable results were obtained from these two observations.

Ultraslow relaxation of hydrogen-bonded dynamic clusters in glass-forming aqueous glucose solutions: A light scattering study

We report static and dynamic light-scattering measurements of aqueous glucose solutions near their glass transition. Photon correlation spectroscopy reveals two relaxation processes present in the supercooled liquid: a nonexponential and nonhydrodynamic, alpha -relaxation occurring at short times and an exponential and hydrodynamic relaxation occurring at longer times. The slow relaxation is seen only in the polarized scattering geometry and is in many ways identical to the "ultraslow" mode recently observed by others in specially annealed molecular glass-forming liquids and attributed to the formation of long-range density correlations or "dynamic clusters." Static light scattering confirms the existence of excess scattering in our glucose solutions that is consistent with clusters in a size range between 30 and 60nm . The size of the clusters varies with the water content and the clustering appears to be associated with the percolation of a hydrogen-bonded glucose network.

Gelation of Native β-Lactoglobulin Induced by Electrostatic Attractive Interaction with Xanthan Gum

The mechanism and kinetics of the electrostatic gelation of native beta-lactoglobulin-xanthan gum mixtures in aqueous solution is reported. The total biopolymer concentration at which gelation was obtained was extremely low (0.1 wt %) compared to the usually tested concentrations for protein-polysaccharide mixed gels (4-12 wt %). This is, to our knowledge, the first time that oppositely charged proteins and polysaccharides are reported to form a gel without applying any treatment to denature the protein (e.g. heating, enzymatic hydrolysis) and at such low concentrations. Static light-scattering and viscoelastic measurements allowed determination of the gelation kinetics. It was found that the gelation process initiated following a similar path as that of an associative phase separation process, i.e., with the formation of primary and interpolymeric electrostatic complexes. However, interpolymeric complexes were able to form clusters and junction zones that resulted in the freeze-in of the whole structure at the point of gelation. The formed gel is therefore a coupled-gel, that is, a gel that has junction zones involving two different molecules. The structuration of xanthan gum, even at these low concentrations, may have played a role in the structuration process. Due to the electrostatic nature of the gels, there was an optimum pH and macromolecular ratio at which the stability of the gels was maximal. This was related to the existence of a stoichiometric electrical charge equivalence pH, where molecules carry equal but opposite charges and protein-polysaccharide interactions are at their maximum.

Scaling analysis of cotton cellulose/LiCl·DMAc solution using light scattering and rheological measurements

AbstractSemidilute solution of cotton lint (CC1) in 8 wt % LiCl/N,N‐dimethylacetamide was investigated using static light scattering (SLS) and rheological measurements. The reduced osmotic modulus estimated by SLS measurements for CC1 solutions are proportional to c1.16 in the semidilute region. From the exponent of 1.16, de Gennes' scaling theory derives the relationship between radius of gyration, Rg, and molecular weight, Mw, of CC1 as Rg ∝ M0.62 This corresponds to the Mark‐Houwink‐Sakurada exponent of 0.86. This exponent is very close to that estimated from scaling analysis of zero shear rate viscosity, that is 0.85. Apparent radius of gyration, Rg,app, estimated by SLS measurements for CC1 solutions are proportional to c−0.5 in the semidilute region. Rg,app indicates the mesh size of polymer entanglement in the semidilute region. On the assumption of the Gaussian behavior of CC1 molecule in the semidilute region, the exponent of −0.5 gives the relationship between the molar mass between entanglements, Me, and c as following relationship: Me ∝ c−1. This agrees with the concentration dependence on plateau modulus estimated from the dynamic viscoelastic measurements. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2155–2160, 2006

Multiple-scattering effects on static light-scattering optical structure factor measurements

We show that the extent and effect of multiple scattering on angularly resolved light-scattering intensity measurements, the optical structure factor, can be quantitatively described by a single parameter, the average number of scattering events along the scattering volume. This quantity is easily measured or calculated and hence provides a useful experimental indicator of multiple scattering, which is a hindrance to accurate structure factor measurements.