Gregory Beaucage

Approximations Leading to a Unified Exponential/Power-Law Approach to Small-Angle Scattering

A new approach to the analysis of small-angle scattering is presented that describes scattering from complex systems that contain multiple levels of related structural features. For example, a mass fractal such as a polymer coil contains two structural levels, the overall radius of gyration and the substructural persistence length. One structural level is described by a Guinier and an associated power-law regime. A function is derived that models both the Guinier exponential and structurally limited power-law regimes without introducing new parameters beyond those used in local fits. Account is made for both a low-q and a high-q limit to power-law scattering regimes. The unified approach can distinguish Guinier regimes buried between two power-law regimes. It is applicable to a wide variety of systems. Fits to data containing multiple power-law and exponential regimes using this approach have previously been reported. Here, arguments leading to the unified approach are given. The usefulness of this approach is demonstrated through comparison with model calculations using the Debye equation for polymer coils (mass fractal), equations for polydisperse spheres (Porod scattering) and randomly oriented ellipsoids of revolution with diffuse interfaces, as well as randomly oriented rod and disc-shaped particles.

Small-Angle Scattering from Polymeric Mass Fractals of Arbitrary Mass-Fractal Dimension

The Debye equation for polymer coils describes scattering from a polymer chain that displays Gaussian statistics. Such a chain is a mass fractal of dimension 2 as evidenced by a power-law decay of −2 in the scattering at intermediate q. At low q, near q ≃ 2π/Rg, the Debye equation describes an exponential decay. For polymer chains that are swollen or slightly collapsed, such as is due to good and poor solvent conditions, deviations from a mass-fractal dimension of 2 are expected. A simple description of scattering from such systems is not possible using the approach of Debye. Integral descriptions have been derived. In this paper, asymptotic expansions of these integral forms are used to describe scattering in the power-law regime. These approximations are used to constrain a unified equation for small-angle scattering. A function suitable for data fitting is obtained that describes polymeric mass fractals of arbitrary mass-fractal dimension. Moreover, this approach is extended to describe structural limits to mass-fractal scaling at the persistence length. The unified equation can be substituted for the Debye equation in the RPA (random phase approximation) description of polymer blends when the mass-fractal dimension of a polymer coil deviates from 2. It is also used to gain new insight into materials not conventionally thought of as polymers, such as nanoporous silica aerogels.

Disposable smart lab on a chip for point-of-care clinical diagnostics

This paper presents the development of a disposable plastic biochip incorporating smart passive microfluidics with embedded on-chip power sources and integrated biosensor array for applications in clinical diagnostics and point-of-care testing. The fully integrated disposable biochip is capable of precise volume control with smart microfluidic manipulation without costly on-chip microfluidic components. The biochip has a unique power source using on-chip pressurized air reservoirs, for microfluidic manipulation, avoiding the need for complex microfluidic pumps. In addition, the disposable plastic biochip has successfully been tested for the measurements of partial oxygen concentration, glucose, and lactate level in human blood using an integrated biosensor array. This paper presents details of the smart passive microfluidic system, the on-chip power source, and the biosensor array together with a detailed discussion of the plastic micromachining techniques used for chip fabrication. A handheld analyzer capable of multiparameter detection of clinically relevant parameters has also been developed to detect the signals from the cartridge type disposable biochip. The handheld analyzer developed in this work is currently the smallest analyzer capable of multiparameter detection for point-of-care testing.

Particle size distributions from small-angle scattering using global scattering functions

Control and quantification of particle size distribution is of importance in the application of nanoscale particles. For this reason, polydispersity in particle size has been the focus of many simulations of particle growth, especially for nanoparticles synthesized from aerosols such as fumed silica, titania and alumina. Single-source aerosols typically result in close to a log-normal distribution in size and micrograph evidence generally supports close to spherical particles, making such particles ideal candidates for considerations of polydispersity. Small-angle X-ray scattering (SAXS) is often used to measure particle size in terms of the radius of gyration, Rg, using Guiniers law, as well as particle surface area, S/V, from the Porod constant B and the scattering invariant Q. In this paper, the unified function is used to obtain these parameters and various moments of the particle size distribution are calculated. The particle size obtained from BET analysis of gas adsorption data directly agrees with the moment calculated from S/V. Scattering results are also compared with TEM particle-counting results. The potential of scattering to distinguish between polydisperse single particles and polydisperse particles in aggregates is presented. A generalized index of polydispersity for symmetric particles, PDI = BRg4/(1.62G), where G is the Guinier prefactor, is introduced and compared with other approaches to describe particle size distributions in SAXS, specifically the maximum-entropy method.

3D Hierarchical orientation in polymer -clay nanocomposite films

Organically modified clay was used as reinforcement for HDPE using maleated polyethylene (PEMA) as a compatibilizer. The effect of compatibilizer concentration on the orientation of various structural features in the polymer-layered silicate nanocomposite (PLSN) system was studied using two-dimensional (2D) small angle X-ray scattering (SAXS) and 2D wide-angle X-ray scattering (WAXS). The dispersion (repeat period) and three-dimensional (3D) orientations of six different structural features were easily identified: (a) clay clusters/tactoids (0.12 mm), (b) modified clay (002) (24 – 31 A u ), (c) unmodified clay (002) (13 A u ), (d) clay (110) and (020) planes normal to (b) and (c), (e) polymer crystalline lamellae (001) (190– 260 A u ), and (f) polymer unit cell (110) and (200) planes. A 3D study of the relative orientation of this hierarchical morphology was carried out by measuring three scattering projections for each sample. Quantitative data on the orientation of these structural units in the nanocomposite film is determined through calculation of the major axis direction cosines and through a ternary, direction-cosine plot. Surprisingly, it is the unmodified clay which shows the most intimate relationship with the polymer crystalline lamellae in terms of orientation. Association between clay and polymer lamellae may be related to an observed increase in lamellar thickness in the composite films. Orientation relationships also reveal that the modified clay is associated with large-scale tactoid structures. q 2002 Elsevier Science Ltd. All rights reserved.

Structural studies of complex systems using small-angle scattering: a unified Guinier/power-law approach☆

A unified analysis method for small-angle scattering data is demonstrated by surveying complex systems that display multiple size-scale structures. Using this approach the relationship between micro- and nano-structures can be ascertained. The method uses a function that is general enough to adequately describe systems ranging from particulates with fractally rough interfaces to mass fractals such as polymer coils. Additionally multiple Guinier and power-law regimes can be treated. The unified method can distinguish Guinier regimes buried between two power-law regimes. Data from particulate filled systems, low crystallinity polymers and low density polymer foams are analyzed.

Rational design of reinforced rubber

Filled elastomer systems have been studied extensively over the past several decades, especially in the application to tire performance. During this time, many attempts have been made to explain reinforcement of an elastomer when fillers are added. These reinforced properties include enhanced strength, modulus, abrasion resistance, and dynamic mechanical properties. Several approaches have been used to separate the contributing influences and to explain how they work. The majority of these approaches look at the structure and property relationships of the fillers and rubbers independently and as a synergistic combination. These approaches have evolved into the following major areas: filler structure, hydrodynamic reinforcement, and interactions involving fillers and elastomers. This paper will review the major works in each of these areas and attempt to offer an overall view of reinforcement of elastomers. Special attention will be paid to the relationships between filler structure and how it may be used to predict reinforcement properties. The general topics that will be covered included filler structure and characterization, rubber and filler interactions, mechanical reinforcement/hydrodynamic effect, and a fractal approach to explaining reinforcement.

Monitoring simultaneously the growth of nanoparticles and aggregates by in situ ultra-small-angle x-ray scattering

Ultra-small-angle x-ray scattering can provide information about primary particles and aggregates from a single scattering experiment. This technique is applied in situ to flame aerosol reactors for monitoring simultaneously the primary particle and aggregate growth dynamics of oxide nanoparticles in a flame. This was enabled through the use of a third generation synchrotron source (Advanced Photon Source, Argonne IL, USA) using specialized scattering instrumentation at the UNICAT facility which is capable of simultaneously measuring nanoscales to microscales (1nmto1μm). More specifically, the evolution of primary-particle diameter, mass-fractal dimension, geometric standard deviation, silica volume fraction, number concentration, radius of gyration of the aggregate, and number of primary particles per aggregate are measured along the flame axis for two different premixed flames. All these particle characteristics were derived from a single and nonintrusive measurement technique. Flame temperature profile...

Structural analysis of poly(dimethylsiloxane) modified silica xerogels

Abstract A new type of inorganic-organic hybrid material incorporating poly(dimethylsiloxane) (PDMS) in tetraethoxysilane (TEOS) based glassy gels has been produced by a sol–gel process. SAXS and SEM studies indicate that these PDMS modified silica xerogels possess multiple size-scale morphologies, ranging from Angstroms to micrometers. The multiple-level hybrid materials show improved structural integrity relative to pure sol–gel glasses (xerogels). The effect of reaction time, water content, as well as content and molecular weight of PDMS on the structure of such materials were studied. These processing variables affect the relative rates of hydrolysis and condensation reactions, and consequently change the microstructure of the final product.

Multilevel structure of reinforcing silica and carbon

Using small-angle x-ray (SAXS), neutron (SANS), x-ray diffraction and light scattering, we study the structure of colloidal silica and carbon on length scales from 4 A < q-1 < 107 A where q is the magnitude of the scattering vector. These materials consist of primary particles of the order of 100 A, aggregated into micron-sized aggregates that in turn are agglomerated into 100 µ agglomerates. The diffraction data show that the primary particles in precipitated silica are composed of highly defective amorphous silica with little intermediate-range order (order on the scale of several bond distances). On the next level of morphology, primary particles arise by a complex nucleation process in which primordial nuclei briefly aggregate into rough particles that subsequently smooth out to become the seeds for the primaries. The primaries aggregate to strongly bonded clusters by a complex process involving kinetic growth, mechanical disintegration and restructuring. Finally, the small-angle scattering (SAS) data lead us to postulate that the aggregates cluster into porous, rough-surfaced, non-mass-fractal agglomerates that can be broken down to the more strongly bonded aggregates by application of shear. We find similar structure in pelletized carbon blacks. In this case we show a linear scaling relation between the primary and aggregate sizes. We attribute the scaling to mechanical processing that deforms the fractal aggregates down to the maximum size able to withstand the compaction stress. Finally, we rationalize the observed structure based on empirical optimization by filler suppliers and some recent theoretical ideas due to Witten, Rubenstein and Colby.