Diffusion on random site percolation clusters. Theory and NMR microscopy experiments with model objects

Quasi two-dimensional random site percolation model objects were fabricate based on computer generated templates. Samples consisting of two compartments, a reservoir of H$_2$O gel attached to a percolation model object which was initially filled with D$_2$O, were examined with NMR (nuclear magnetic resonance) microscopy for rendering proton spin density maps. The propagating proton/deuteron inter-diffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters. The deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster. The fractal dimension of the random walk, $d_w$, evaluated from the diffusion measurements on the one hand and the fractal dimension, $d_f$, deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents. Approximate calculations of the propagator are given on the basis of the fractional diffusion equation. Furthermore, the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison. The anomalous diffusion constant was evaluated and is compared to the Brownian case. Some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system. In this way, anomalous solutions of the fractional diffusion equation could experimentally be verified for the first time.Comment: REVTeX, 12 figures in GIF forma

Associations of Maternal Complaints to Levator Ani Muscle Trauma within 9 Months after Vaginal Birth: A Prospective Observational Cohort Study

Introduction: Pelvic floor trauma in the form of partial or complete avulsions of the levator ani muscle (LAM) affects 6-42% of women after vaginal birth and can cause tremendous long-term morbidity. Many studies assessed morphological pelvic floor trauma after childbirth but lacked to evaluate women's associated short-term complaints. A proper assessment of trauma and subjective complaints after birth could help to assess possible associations between them and their relevance to women's daily life. Therefore, we aimed to assess women's complaints within the first months after birth in association to their LAM trauma. Materials and methods: Between 3/2017 and 4/2019, we prospectively evaluated vaginal births of 212 primiparous women with singletons in vertex presentation ≥ 36 + 0 gestational weeks for levator ani muscle (LAM) trauma by translabial ultrasound, for pelvic organ prolapse by clinical examination, and for urogynecological complaints using questionnaires 1-4 days (P1), 6-10 weeks (P2), and 6-9 months (P3) after birth. The questionnaires were self-designed but oriented to and modified from validated questionnaires. Women's complaints were evaluated for P1-P3 according to their LAM trauma state. Results: At P1, 67% of women showed an intact LAM, whereas 14.6% presented a hematoma, 6.6% a partial avulsion (PAV), and 11.8% a complete avulsion (CAV). At P2, 75.9% showed an intact LAM, 9.9% a PAV, and 14.2% a CAV. At P3, 72.9% of women with a LAM trauma in P1 and/or P2 were assessed with 21.6% being intact and 39.2% having a PAV and CAV, respectively. Obstetrical and baseline characteristics differed slightly between the groups. When comparing the time before and during pregnancy with the time after childbirth, birth itself affected women's complaints in all LAM state groups, but the presence of a LAM trauma, especially a CAV, had more negative effects. Conclusions: Vaginal birth changes the anatomical structure of the maternal birth canal and genital tract, and it alters women's perceptions and body function. In our study, LAM trauma did not change these effects tremendously within the first months. Therefore, other maternal, fetal, and obstetrical factors need consideration for the explanation of maternal complaints, in addition to long-term effects of trauma and dysfunction of the LAM and other birth canal structures

The intra- and intermolecular basis of the zero-shear viscosity in unentangled polymers

A general expression for the zero-shear viscosity in unentangled polymers is derived on the basis of chain pair correlations of normal modes and mutual interaction forces. © 1999 American Institute of Physics

Length and time scales of entanglement and confinement effects constraining polymer chain dynamics

With characteristic time constants for polymer dynamics, namely τs (the segment fluctuation time), τe (the entanglement time), and τR (the longest Rouse relaxation time), the time scales of particular interest (i) τ < τss (ii) τS < t < τe, and (iii) τe < t < τR will be discussed and compared with experimental data. These ranges correspond to the chain-mode length scales (i) ℓ < b, (ii) b < ℓ < d2 /b, and (iii) d2/b < ℓ < L, where b is the statistical segment length, d is the dimension of constraints by entanglements and/or confinement, and L is the chain contour length. Based on Langevin-type equations-of-motion coarse-grained predictions for the mean-squared segment displacement and the spin-lattice relaxation dispersion will be outlined for the scenarios "freely-draining", "entangled", and "confined". In the discussion we will juxtapose "local" versus "global" dynamics on the one hand, and "bulk" versus "confined" systems on the other. © 2010 Materials Research Society

Self-diffusion studies by intra- and inter-molecular spin-lattice relaxometry using field-cycling: Liquids, plastic crystals, porous media, and polymer segments

© 2017 Elsevier B.V.Field-cycling NMR relaxometry is a well-established technique for probing molecular dynamics in a frequency range from typically a few kHz up to several tens of MHz. For the interpretation of relaxometry data, it is quite often assumed that the spin-lattice relaxation process is of an intra-molecular nature so that rotational fluctuations dominate. However, dipolar interactions as the main type of couplings between protons and other dipolar species without quadrupole moments can imply appreciable inter-molecular contributions. These fluctuate due to translational displacements and to a lesser degree also by rotational reorientations in the short-range limit. The analysis of the inter-molecular proton spin-lattice relaxation rate thus permits one to evaluate self-diffusion variables such as the diffusion coefficient or the mean square displacement on a time scale from nanoseconds to several hundreds of microseconds. Numerous applications to solvents, plastic crystals and polymers will be reviewed. The technique is of particular interest for polymer dynamics since inter-molecular spin-lattice relaxation diffusometry bridges the time scales of quasi-elastic neutron scattering and field-gradient NMR diffusometry. This is just the range where model-specific intra-coil mechanisms are assumed to occur. They are expected to reveal themselves by characteristic power laws for the time-dependence of the mean-square segment displacement. These can be favorably tested on this basis. Results reported in the literature will be compared with theoretical predictions. On the other hand, there is a second way for translational diffusion phenomena to affect the spin-lattice relaxation dispersion. If rotational diffusion of molecules is restricted, translational diffusion properties can be deduced even from molecular reorientation dynamics detected by intra-molecular spin-lattice relaxation. This sort of scenario will be relevant for adsorbates on surfaces or polymer segments under entanglement and chain connectivity constraints. Under such conditions, reorientations will be correlated with translational displacements leading to the so-called RMTD relaxation process (reorientation mediated by translational displacements). Applications to porous glasses, protein solutions, lipid bilayers, and clays will be discussed. Finally, we will address the intriguing fact that the various time limits of the segment mean-square displacement of polymers in some cases perfectly reproduce predictions of the tube/reptation model whereas the reorientation dynamics suggests strongly deviating power laws

Theory of field-gradient NMR diffusometry of polymer segment displacements in the tube-reptation model

The spin-echo attenuation in NMR field-gradient diffusometry experiments is treated for the tube model in a time scale longer than the entanglement time e. The theory comprises the Doi-Edwards [M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)] limits of the (anomalous) segment displacement as well as the (ordinary) center-of-mass diffusion. This formalism is to be distinguished from formalisms for anomalous diffusion on fractal networks: The reptation mechanism implies an intrinsically different character of the displacement probability density. It is shown that the expressions usually applied in NMR diffusometry are inadequate for the reptation problem and can cause misinterpretations. Applications of the formalism to polymer chains in bulk and confined in porous media are discussed. © 1995 The American Physical Society

Anisotropy of the segment mobility versus self-and pair-Correlation functions in polymer melts under mesoscopic confinement

Techniques typically used for studies of polymer dynamics such as NMR relaxometry, quasielastic neutron scattering, multiple-quantum build-up NMR, dielectric relaxation spectroscopy, and mechanical relaxation are specified and commonly classified in terms of correlation functions. Two categories of correlation functions are identified with respect to their specific ability to describe translational fluctuations on the one hand, and molecular reorientations on the other. The first category is of the dynamic-structure factor type reflecting the absolute or relative displacement behavior of particles. This type of function is in contrast to the second category, namely correlation functions of spherical harmonics of different orders characterizing rotational diffusion of molecules or molecular groups. In polymers, rotational diffusion tends to be strongly anisotropic. It is elucidated that the long-time tail of the correlation decay is particularly indicative for model characteristic features. The representation of experimental results by correlation functions instead of method- specific technical terms permits unambiguous comparisons and interpretations based on different techniques. This is demonstrated in the context of a problem of particularly topical interest, namely polymer melts confined in nanoscopic porous matrices. Methods probing correlation functions of spherical harmonics are shown to be sensitive to rotational chain dynamics severely modified under geometrical confinement, the so-called corset effect. On the other hand, correlation functions of the dynamical structure factor type characterizing translational fluctuations reveal little influence of such constraints in the experimentally accessible time/space window. In order to support this classification, the potentially competitive influence of wall adsorption effects is discussed in addition. Criteria permitting to rule out this sort of retardation mechanism under appropriate conditions are specified. © 2010 American Chemical Society

Nuclear spin-lattice relaxation dispersion and segment diffusion in entangled polymers. Renormalized Rouse formalism

A formalism for polymer melts was derived linking the spin-lattice relaxation time T1, the correlation function of chain tangent vectors and the mean-square segment displacement with memory functions. Potential normal-mode number dependences are included. In the limit of infinitely fast decaying memory functions the theory reproduces known expressions characteristic for Rouse dynamics. Interchain excluded-volume forces were taken into account in the frame of the renormalized Rouse approach [K. S. Schweizer, J. Chem. Phys. 91, 5802 (1989)]. The power law limits predicted on this basis are T 1, ∝ω1/2, T1∝ω1/4, and T1∝ω1/5 for the T1 dispersion in a sequence of regimes from high to low frequencies. The mean-square segment displacement obeys 〈r2〉∝t1/4, 〈r2〉∝ t3/8, and 〈r2〉∝2/5 in a sequence of limits for increasing times. The spin-lattice relaxation dispersion of different polymers was studied mainly by the aid of the field-cycling NMR technique. The covered proton frequency range is less than 103 Hz to more than 108 Hz. The frequency dependence can be described by a series of power laws arising from chain dynamics. Two of these, namely T 1∝ω0.5 and T1∝ω0.25 tending to appear at high and low frequencies, respectively, can be perfectly explained on the basis of the derived renormalized Rouse limits. The third power law, T1∝ω0.44, which was observed only at rather low frequencies, has no theoretical counterpart in the frame of the renormalized Rouse theory. Some hints that farther reaching polymer theories such as the mode-mode coupling approach [K. S. Schweizer, J. Chem. Phys. 91, 5822 (1989)] can help to understand this finding are discussed. © 1994 American Institute of Physics

Length and time scales of entanglement and confinement effects constraining polymer chain dynamics

With characteristic time constants for polymer dynamics, namely τs (the segment fluctuation time), τe (the entanglement time), and τR (the longest Rouse relaxation time), the time scales of particular interest (i) τ < τss (ii) τS < t < τe, and (iii) τe < t < τR will be discussed and compared with experimental data. These ranges correspond to the chain-mode length scales (i) ℓ < b, (ii) b < ℓ < d2 /b, and (iii) d2/b < ℓ < L, where b is the statistical segment length, d is the dimension of constraints by entanglements and/or confinement, and L is the chain contour length. Based on Langevin-type equations-of-motion coarse-grained predictions for the mean-squared segment displacement and the spin-lattice relaxation dispersion will be outlined for the scenarios "freely-draining", "entangled", and "confined". In the discussion we will juxtapose "local" versus "global" dynamics on the one hand, and "bulk" versus "confined" systems on the other. © 2010 Materials Research Society

Critical fluctuations and random-anisotropy glass transition in nematic elastomers

We carry out a detailed deuterium NMR study of local nematic ordering in polydomain nematic elastomers. This system has a close analogy to the random-anisotropy spin glass. We find that, in spite of the quadrupolar nematic symmetry in 3-dimensions requiring a first-order transition, the order parameter in the quenched ``nematic glass'' emerges via a continuous phase transition. In addition, by a careful analysis of the NMR line shape, we deduce that the local director fluctuations grow in a critical manner around the transition point. This could be the experimental evidence for the Aizenman-Wehr theorem about the quenched impurities changing the order of discontinuous transition