Time-dependent Parts Research Articles

Observations of the coil—globule transition in single polymers

Abstract We report observations of the coil-globule transition of a single polymer chain in solution. For polyacrylamide (MW = 5−6×106), the transition occurs when the acetone concentration of the acetone water solvent is increased. In polystyrene (MW = 26×106)—cyclohexane solutions, the transition is induced by lowering the temperature. Since the transition to the globule state is in competition with inter-polymer aggregation, measurements must be made at very low polymer concentrations. By measuring the angular dependence of the time-dependent and time-independent parts of the correlation function of light scattered from dilute solutions of polymer (<.1 μg/m1), we have determined the radius of gyration and hydordynamic radius of the globule state before the onset of aggregation.

Data reduction methodology for perturbed angular correlation experiments

Data reduction schemes currently used for time differential perturbed angular correlation measurements are evaluated in terms of (i) their relative effectiveness in eliminating irrelevant variables, including single counter efficiencies and spectrum time shifts; and (ii) their effectiveness in putting data in a form that can easily be fitted by theoretical correlation functions. It is pointed out that erroneous conclusions may be derived from improperly reduced data, but that properly analyzed experiments performed at four angles allow a good determination of both the time-dependent and time-independent parts of the correlation function. Correction of data for accidentals, source self-absorption and spectrum livetime differences are discussed.

Transient behaviour of porous electrodes with high exchange current densities

A theoretical analysis of the non-steady-state reaction distribution in a porous electrode with a high exchange current density is made by application of a simplified macrohomogeneous model for porous electrodes. The dimensionless transfer current for short times is given as an expansion in time, and two terms in a moderate time solution are also presented. At moderate times the current is split into time-dependent and time-independent parts, but this distinction is not apparent in the short time solution. For the limiting case of reversible kinetics, the analysis specifies the fraction of the superficial current density that will be distributed through the electrode. Treatments for the PbO 2 and LiAl electrodes are presented as examples.

The photoelastic behaviour of swollen networks of polymethacrylic acid

The visoelastic photoelastic behaviour of cross-linked polymethacrylic acid, swollen to equilibrium in variously concentrated aqueous NaCl solutions ( c x = 0.05-1.5 M), was studied. In the rubber-like region, stress and birefringence relaxation at various elongations (λ) can be separated into the deformation- and time-dependent parts. Both the extrapolated equilibrium modulus ( G r ) and the interaction parameter (χ) strongly depend on the degree of swelling of the gel. One of the most interesting findings is that the exponent of the volume fraction of the polymer ( ν 2) in the dependence of the modulus G e on ν 2 is much higher than 1/3 as predicted by the theory of Gaussian networks, if the concentration of the crosslinking agent c is low. The same is true for the Mooney-Rivlin constant ( C 1) even if the ratio C 2 C 1 approaches zero at lower ν 2. The values of the stress-optical coefficient ( C e ) increase with both increasing swelling and decreasing content of the cross-linking agent in the gel; the most pronounced increase in C e is connected with the largest decrease in G e . The same dependence of C e on the modulus in the dry state ( G d ), irrespective of the concentration of the cross-linking agent and c s , suggests the conclusive role played by the macroform contribution to birefringence. The effects found are probably due to the disappearance of the secondary structure of PMA chains with increasing swelling and concentration of the cross-linking agent.

Neutron Counting Statistics in the Family-First-Pulse-Triggered Gate, (II)

The first moment of neutron counts registered in the family-first-pulse-triggered gate—i.e. the object of observation in the usual Rossi-a experiment—is analyzed as a function of waiting time from the instant of family-birth to that of the trigger. The correlation amplitude is shown to increase with the waiting time, yielding when averaged over the waiting time distribution the currently used expression of this amplitude. The structure of the uncorrelated count-rate is made clear, and is shown to consist of two kinds of time-dependent parts, which are formulated as functions of time after the trigger.

Neutron Counting Statistics in the Family-First-Pulse-Triggered Gate, (I)

The first moment of neutron counts registered in the family-first-pulse-triggered gate—i.e. the object of observation in the usual Rossi-a experiment—is analyzed as a function of waiting time from the instant of family-birth to that of the trigger. The correlation amplitude is shown to increase with the waiting time, yielding when averaged over the waiting time distribution the currently used expression of this amplitude. The structure of the uncorrelated count-rate is made clear, and is shown to consist of two kinds of time-dependent parts, which are formulated as functions of time after the trigger.

Effect of anisotropic turbulence on aerodynamic noise

A model based on Lighthill's theory for predicting aerodynamic noise from a turbulent shear flow is developed. This model is a generalization of the one developed by Ribner. It does not require that the turbulent correlations factor into space and time-dependent parts. It replaces his assumption of isotropic turbulence by the more realistic one of axisymmetric turbulence. In the course of the analysis, a hierarchy of equations is developed wherein each succeeding equation involves more assumptions than the preceding equation but requires less experimental information for its use. The implications of the model for jet noise are discussed. It is shown that for the particular turbulence data considered anisotropy causes the high-frequency self-noise to be beamed downstream.

Convective Spherical Shell: II. With Rotation

Abstract The problem of a convective, rotating spherical shell is considered in Herring's approximation. The temperature is expanded in spherical harmonics, YLm(θ,Φ), and the velocity field in basic poloidal pLm(r,t) and toroidal tLm(r,t) vectors. The scalar pLm(r,t) [tLm(r,t)], together with YLm(θ,Φ), defines a basic poloidal [toroidal] vector. The equations for pLm(r,t) and tLm(r,t) with different L's are coupled by the Taylor number and two types of solutions are possible: symmetric or antisymmetric about the equator. For the case of axial symmetry and for a Rayleigh number equal to 1500, we calculate the convective steady-state solution with rotation by successively increasing the Taylor number from zero, its value for no rotation. Using free surface boundary conditions, the relevant equations determine the radial and time-dependent parts of the temperature and velocity field, with the exception of t10(r), the lowest toroidal component of the axisymmetric solution having equatorial symmetry. The conse...

Nuclear Magnetic Resonance Saturation and Rotary Saturation in Solids

Nuclear spin-lattice relaxation times of ${\mathrm{Al}}^{27}$ in pure Al and ${\mathrm{Cu}}^{63}$ in annealed pure Cu have been measured with a nuclear induction spectrometer, by the method of saturation. The experimental values of ${T}_{1}$ are 4.1\ifmmode\pm\else\textpm\fi{}0.8 milliseconds for ${\mathrm{Al}}^{27}$ and 3.0\ifmmode\pm\else\textpm\fi{}0.6 milliseconds for ${\mathrm{Cu}}^{63}$, in reasonable agreement with theory.The dispersion mode of the nuclear resonance was also observed, and it was found that ${\ensuremath{\chi}}^{\ensuremath{'}}$ (the real part of the rf susceptibility) does not saturate at the same level as the absorption, ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$, but remains roughly constant out to a radio-frequency field intensity of about 2 gauss. Both ${\ensuremath{\chi}}^{\ensuremath{'}}$ and ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$ become narrower and nearly Lorentzian in shape above saturation. When the dc magnetic field modulation is increased from 14 to 41 cps the phase of the dispersion signal lags behind the modulation, presumably because the modulation period is then comparable to ${T}_{1}$. Large dispersion signals above have also been observed for the ${\mathrm{Na}}^{23}$ resonance in NaCl.This behavior of the dispersion mode is in conflict with the predictions of Bloembergen, Purcell, and Pound and of the Bloch equations. The validity of these theories is re-examined, and it is concluded that although they are applicable to nuclear resonance in liquids and gases, and to solids at small rf intensities, they contain incorrect assumptions as applied to solids at high rf power levels. The theory of Bloembergen, Purcell, and Pound is based on an assumption equivalent to that of a spin temperature. It is shown that the spin state cannot be strictly described by a spin temperature because the phases of the spin quantum states are not incoherent, as required by the temperature concept. The transverse decay of the nuclear magnetization predicted by the Bloch equations is shown to be partially forbidden by energy and entropy considerations if a large rf field at the resonance frequency is continuously applied to the solid.A theory is developed which is applicable only to solids at rf magnetic field intensities well above the level and which is in reasonable agreement with the experimental observations. The Hamiltonian is transformed to a coordinate system rotating at the frequency of the rf field. The resulting time-dependent parts of the spin-spin interaction are nonsecular perturbations on the time-independent part, and can therefore be ignored. Statistical mechanics is applied to the remaining stationary spin Hamiltonian; specifically it is assumed that the spin system is in its most probable macrostate (a canonical distribution of quantum states) with respect to the transformed spin Hamiltonian. This assumption is justified because the transformed spin Hamiltonian is effectively time independent and the spin-lattice interaction is small, and it is analogous to assumptions basic to classical acoustics and fluid mechanics. The spin-lattice interaction merely determines the expectation value of the transformed spin Hamiltonian, which can be readily calculated under the assumption that the expectation value of the spin angular momentum of each spin is relaxed independently to its thermal equilibrium value by the lattice in time ${T}_{1}$. Both fast and slow modulation of the dc magnetic field can be treated.Rotary saturation is observed by applying an audio-frequency magnetic field to the sample in the dc field direction while observing the dispersion derivative at resonance with a large rf field ${H}_{1}$. When the audio-frequency approaches $\ensuremath{\gamma}{H}_{1}$ the dispersion signal decreases and goes through a minimum. The effect is easily treated theoretically in solids, liquids and gases by using a rotating coordinate system, and is a rotary analogue of ordinary saturation. It is a convenient method for calibrating rf magnetic fields and appears potentially capable of providing useful information on the solid state. Experimental data on rotary are presented and discussed.