Richard J. Braun

Thermodynamically-consistent phase-field models for solidification

In an effort to unify the various phase-field models that have been used to study solidification, we have developed a class of phase-field models for crystallization of a pure substance from its melt. These models are based on an entropy functional, as in the treatment of Penrose and Fife, and are therefore thermodynamically consistent inasmuch as they guarantee spatially local positive entropy production. General conditions are developed to ensure that the phase field takes on constant values in the bulk phases. Specific forms of a phase-field function are chosen to produce two models that bear strong resemblances to the models proposed by Langer and Kobayashi. Our models contain additional nonlinear functions of the phase field that are necessary to guarantee thermodynamic consistency.

The Contribution of Lipid Layer Movement to Tear Film Thinning and Breakup

PURPOSE To investigate whether the tear film thinning between blinks is caused by evaporation or by tangential flow of the tear film along the surface of the cornea. Tangential flow was studied by measuring the movement of the lipid layer. METHODS Four video recordings of the lipid layer of the tear film were made from 16 normal subjects, with the subjects keeping their eyes open for up to 30 seconds after a blink. To assess vertical and horizontal stretching of the lipid layer and underlying aqueous layer, lipid movement was analyzed at five positions, a middle position 1 mm below the corneal center, and four positions respectively 1 mm above, below, nasal, and temporal to this middle position. In addition, in 13 subjects, the thinning of the tear film after a blink was measured. RESULTS The total upward movement could be fitted by the sum of an exponential decay plus a slow steady drift; this drift was upward in 14 of 16 subjects (P = 0.002). Areas of thick lipid were seen to expand causing upward or downward drift or horizontal movement. The velocity of the initial rapid upward movement and the time constant of upward movement were found to correlate significantly with tear film thickness but not with tear-thinning rate. CONCLUSIONS Analysis indicated that the observed movement of the lipid layer was too slow to explain the observed thinning rate of the tear film. In the Appendix, it is shown that flow under a stationary lipid layer cannot explain the observed thinning rate. It is concluded that most of the observed tear thinning between blinks is due to evaporation.

Contributions of evaporation and other mechanisms to tear film thinning and break-up.

Purpose. To evaluate the contribution of three mechanisms—evaporation of the tear film, inward flow of water into the corneal epithelium or contact lens, and “tangential flow” along the surface of epithelium or contact lens—to the thinning of the tear film between blinks and to tear film break-up. In addition to a discussion of relevant studies, some previously unpublished images are presented illustrating aspects of tear film break-up. Contributions of Three Mechanisms to Tear Film Break-Up. Inward flow of water into the epithelium or contact lens is probably unimportant, and a small flow in the opposite direction may actually occur. Tangential flow is probably important in certain special cases of tear film break-up—at the black line near the tear meniscus, over surface elevations, after partial blinks, and from small thick lipid spots in the tear film. In all these special cases it is argued that tangential flow is important initially, but evaporation may be needed for final thinning to break-up. It is argued that most of the observed tear film thinning between blinks is due to evaporation, rather than tangential flow, and that large “pool” break-up regions are the result of evaporation over an extended area. Conclusion. Evaporation in our “free-air” conditions may be four to five times faster than the average of the values reported in the literature when air currents are prevented by preocular chambers. However, recent evaporation measurements using “ventilated chambers” give higher values, which may correspond better to free-air conditions. Thus evaporation may be fast enough to explain many cases of tear film break-up, and to give rise to considerable increases in the local osmolarity of the tear film between blinks.

Adaptive phase-field computations of dendritic crystal growth

Phase-field models have been the subject of much interest in the last several years for the investigation of phase transitions with complicated morphology, such as dendritic growth. The phase-field method introduces a continuous transition between the two phases across a thin layer of finite thickness; the advantage of this approach is that the location of the interface does not have to be explicitly determined as part of the solution but is obtained from the solution of an additonal field equation representing the evolution of the phase-field variable over the entire domain. A brief overview is presented of the phase-field model development for a pure material using irreversible thermodynamics. The computational model includes four-fold anisotropy both in surface energy and interfacial kinetics. Numerical solutions are obtained using a general-purpose adaptive finite-difference algorithm. Adaptivity in space and time is found to extend somewhat the parameter regime where computations can be carried out. Good convergence to sharp-interface models is achieved for dimensionless undercoolings of 0.25, but a relatively small amount of solid phase grows before the thermal field is affected by the size of the computational domain. Further progress to smaller undercoolings will have to be aided by more sophisticated modeling.

Anisotropy of interfaces in an ordered alloy: a multiple–order–parameter model

A multiple–order–parameter theory of ordering on a binary face–centred–cubic (FCC) crystal lattice is developed, and adapted to provide a continuum formulation that incorporates the underlying symmetries of the FCC crystal in both the bulk and gradient–energy terms of the free energy. The theory is used to compute the orientation dependence of the structure and energy of interphase and antiphase boundaries. The structure of these interfaces compares favourably with previous lattice calculations by Kikuchi and Cahn (1962, 1979). Anisotropy is a natural consequence of the lattice calculation and the multiple–order–parameter continuum formulation presented here. This is in contrast to the ad hoc fashion in which anisotropy is often introduced into a single–order–parameter continuum theory.

Lubrication theory for reactive spreading of a thin drop

Solder drops spreading on metallic substrates are a reactive form of the wetting problem. A metallic component may diffuse in the liquid toward a metal substrate, where it is consumed by a reaction that forms a solid intermetallic phase. The resulting spatial variation in the composition of the drop may cause composition gradients along the free surface of the drop. Together with any thermal gradients along the free surface, Marangoni effects may, in turn, modify the bulk transport in the spreading drop. Motivated by this situation, we extend lubrication theory for the spreading of thin drops in the presence of gravity and thermocapillarity to include mass transport and solutocapillarity. We use an approximate solute profile in the drop to derive coupled evolution equations for the free surface shape and concentration field. Numerical solutions for the nonreactive (single component) drop agree well with previous theory. In the reactive case, we are only able to compute results for parameters outside of th...

High resolution microscopy of the lipid layer of the tear film.

Tear film evaporation is controlled by the lipid layer and is an important factor in dry eye conditions. Because the barrier to evaporation depends on the structure of the lipid layer, a high resolution microscope has been constructed to study the lipid layer in dry and in normal eyes. The microscope incorporates the following features. First, a long working distance microscope objective is used with a high numerical aperture and resolution. Second, because such a high resolution objective has limited depth of focus, 2000 images are recorded with a video camera over a 20-sec period, with the expectation that some images will be in focus. Third, illumination is from a stroboscopic light source having a brief flash duration, to avoid blurring from movement of the lipid layer. Fourth, the image is in focus when the edge of the image is sharp - this feature is used to select images in good focus. Fifth, an aid is included to help align the cornea at normal incidence to the axis of the objective so that the whole lipid image can be in focus. High resolution microscopy has the potential to elucidate several characteristics of the normal and abnormal lipid layer, including different objects and backgrounds, changes in the blink cycle, stability and fluidity, dewetting, gel-like properties and possible relation to lipid domains. It is expected that high resolution microscopy of the lipid layer will provide information about the mechanisms of dry eye disorders. Illustrative results are presented, derived from over 10,000 images from 375 subjects.

Model problems for the tear film in a blink cycle: single-equation models

We consider model problems for the tear film over multiple blink cycles in limits that yield a single equation for the tear film; the single nonlinear partial differential equation that governs the film thickness arises from lubrication theory. The two models arise from considering the absence of naturally occurring surfactant and the case when the surfactant strongly affects the surface tension. The film is considered on a sinusoidally varying domain length with specified film thickness and volume flux at each end; only one end of the domain is moving, which is analogous to the upper eyelid moving with each blink. A main contribution of this article is computation of solutions for multiple complete blink cycles; the results of these non-trivial computations show a distinct similarity to quantitative in vivo observations of the tear film under partial blink conditions. A transition between periodic and non-periodic solutions has been estimated and this may be a criterion for what is effectively a full blink according to fluid dynamic considerations.

Tear film breakup and structure studied by simultaneous video recording of fluorescence and tear film lipid layer images.

PURPOSE The thinning of the precorneal tear film between blinks and tear film breakup can be logically analyzed into contributions from three components: evaporation, flow into the cornea, and tangential flow along the corneal surface. Whereas divergent tangential flow contributes to certain types of breakup, it has been argued that evaporation is the main cause of tear thinning and breakup. Because evaporation is controlled by the tear film lipid layer (TFLL) it should therefore be expected that patterns of breakup should match patterns in the TFLL, and this hypothesis is tested in this study. METHODS An optical system is described for simultaneous video imaging of fluorescein tear film breakup and the TFLL. Recordings were made from 85 subjects, including both with healthy and dry eyes. After instillation of 5 μL2% fluorescein, subjects were asked to blink 1 second after the start of the recording and try to maintain their eyes open for the recording length of 30 or 60 seconds. RESULTS Areas of tear film thinning and breakup usually matched corresponding features in the TFLL. Whereas thinning and breakup were often matched to thin lipid, surprisingly, the corresponding lipid region was not always thinner than the surrounding lipid. Occasionally, a thin lipid region caused a corresponding region of greater fluorescence (thicker aqueous layer), due to convergent tangential flow. CONCLUSIONS Areas of tear thinning and breakup can generally be matched to corresponding regions of the TFLL as would be expected if breakup is largely due to evaporation. Surprisingly, in some examples, the corresponding lipid area was not thinner and possibly thicker than the surrounding lipid. This indicates that the lipid was a poor barrier to evaporation, perhaps because of deficiency in composition and/or structure. For example, bacterial lipases may have broken down esters into component acids and alcohols, causing a defective TFLL structure with increased evaporation.

Tear film dynamics on an eye-shaped domain. Part 2. Flux boundary conditions

We model the dynamics of the human tear film during relaxation (after a blink) using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the flux of tear fluid into or out of the domain. The governing nonlinear partial differential equation is solved on an overset grid by a method of lines using finite differences in space and an adaptive second-order backward difference formula solver in time. Our simulations in a two-dimensional domain are computed in the Overture computational framework. The flow around the boundary is sensitive to both our choice of flux boundary condition and the presence of gravity. The simulations recover features seen in one-dimensional simulations and capture some experimental observations of tear film dynamics around the lid margins. In some instances, the influx from the lacrimal gland splits with some fluid going along the upper lid towards the nasal canthus and some travelling around the temporal canthus and then along the lower lid. Tear supply can also push through some parts of the black line near the eyelid margins.