Spreading of liquid drops over saturated porous layers

Spreading of small liquid drops over thin porous layers saturated with the same liquid is investigated from both theoretical and experimental points of view. A theory is presented that shows that spreading is governed by the same power law as in the case of spreading over a dry solid substrate. The Brinkman's equations are used to model the liquid flow inside the porous substrate. An equation of the drop spreading is deduced, which shows that both an effective lubrication and the liquid exchange between the drop and the porous substrates are equally important. The presence of these two phenomena removes the well-known singularity at the moving three-phase contact line. Matching of the drop profile in the vicinity of the three-phase contact line with the main spherical part of the drop gives the possibility to calculate the pre-exponential factor in the spreading law via permeability and effective viscosity of the liquid in the porous layer. Unfortunately, the latter dependency turns out to be very weak. Spreading of silicone oils over different microfiltration membranes is carried out. Radii of spreading on time experimental dependencies confirm the theory predictions. Experimentally found coefficients agree with theoretical estimations.     

Dependence of volatile droplet lifetime on the hydrophobicity of the substrate

In this Article, we demonstrate the dependence of the lifetime of a volatile droplet on the hydrophobicity of the substrate. Ethanol droplets placed on the molecularly smooth surfaces of three polymers, applied to substrates by spin-coating, showed distinct types of behavior depending on the hydrophobicity of the latter. High contact angles, θ, lead to fairly regular recession of the triple line during liquid evaporation at essentially constant θ, whereas low contact angle caused pinning, θ decreasing with time. The latter case leads to shorter drop lifetimes.     

Surfactant solutions and porous substrates: spreading and imbibition

In Section 1, spreading of small liquid drops over thin dry porous layers is investigated from both theoretical and experimental points of view [V.M. Starov, S.R. Kosvintsev, V.D. Sobolev, M.G. Velarde, S.A. Zhdanov, J. Colloid Interface Sci. 252 (2002) 397]. Drop motion over a porous layer is caused by an interplay of two processes: (a) the spreading of the drop over already saturated parts of the porous layer, which results in an expanding of the drop base, and (b) the imbibition of the liquid from the drop into the porous substrate, which results in a shrinkage of the drop base and an expanding of the wetted region inside the porous layer. As a result of these two competing processes, the radius of the drop goes through a maximum value over time. A system of two differential equations has been derived to describe the evolution with time of radii of both the drop base and the wetted region inside the porous layer. This system includes two parameters, one accounts for the effective lubrication coefficient of the liquid over the wetted porous substrate, and the other is a combination of permeability and effective capillary pressure inside the porous layer. Two additional experiments were used for an independent determination of these two parameters. The system of differential equations does not include any fitting parameter after these two parameters are determined. Experiments were carried out on the spreading of silicone oil drops over various dry microfiltration membranes (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region inside the porous layer were monitored. All experimental data fell on two universal curves if appropriate scales are used with a plot of the dimensionless radii of the drop base and of the wetted region inside the porous layer on dimensionless time. The predicted theoretical relationships are two universal curves accounting quite satisfactory for the experimental data. According to theory predictions [1]: (i) the dynamic contact angle dependence on the same dimensionless time as before should be a universal function, and (ii) the dynamic contact angle should change rapidly over an initial short stage of spreading and should remain a constant value over the duration of the rest of the spreading process. The constancy of the contact angle on this stage has nothing to do with hysteresis of the contact angle: there is no hysteresis in the system under investigation. These conclusions again are in good agreement with experimental observations [V.M. Starov, S.R. Kosvintsev, V.D. Sobolev, M.G. Velarde, S.A. Zhdanov, J. Colloid Interface Sci. 252 (2002) 397]. In Section 2, experimental investigations are reviewed on the spreading of small drops of aqueous SDS solutions over dry thin porous substrates (nitrocellulose membranes) in the case of partial wetting [S. Zhdanov, V. Starov, V. Sobolev, M. Velarde, Spreading of aqueous SDS solutions over nitrocellulose membranes. J. Colloid Interface Sci. 264 (2003) 481-489]. The time evolution was monitored of the radii of both the drop base and the wetted area inside the porous substrate. The total duration of the spreading process was subdivided into three stages-the first stage: the drop base expands until the maximum value of the drop base is reached; the contact angle rapidly decreases during this stage; the second stage: the radius of the drop base remains constant and the contact angle decreases linearly with time; the third stage: the drop base shrinks and the contact angle remains constant. The wetted area inside the porous substrate expends during the whole spreading process. Appropriate scales were used with a plot of the dimensionless radii of the drop base, of the wetted area inside the porous substrate, and the dynamic contact angle on the dimensionless time. Experimental data showed [S. Zhdanov, V. Starov, V. Sobolev, M. Velarde, Spreading of aqueous SDS solutions over nitrocellulose membranes. J. Colloid Interface Sci. 264 (2003) 481-489]: the overall time of the spreading of drops of SDS solution over dry thin porous substrates decreases with the increase of surfactant concentration; the difference between advancing and hydrodynamic receding contact angles decreases with the surfactant concentration increase; the constancy of the contact angle during the third stage of spreading has nothing to do with the hysteresis of contact angle, but determined by the hydrodynamic reasons. It is shown using independent spreading experiments of the same drops on nonporous nitrocellulose substrate that the static receding contact angle is equal to zero, which supports the conclusion on the hydrodynamic nature of the hydrodynamic receding contact angle on porous substrates. In Section 3, a theory is developed to describe a spontaneous imbibition of surfactant solutions into hydrophobic capillaries, which takes into account the micelle disintegration and the concentration decreasing close to the moving meniscus as a result of adsorption, as well as the surface diffusion of surfactant molecules [N.V. Churaev, G.A. Martynov, V.M. Starov, Z.M. Zorin, Colloid Polym. Sci. 259 (1981) 747]. The theory predictions are in good agreement with the experimental investigations on the spontaneous imbibition of the nonionic aqueous surfactant solution, Syntamide-5, into hydrophobized quartz capillaries. A theory of the spontaneous capillary rise of surfactant solutions in hydrophobic capillaries is presented, which connects the experimental observations with the adsorption of surfactant molecules in front of the moving meniscus on the bare hydrophobic interface [V.J. Starov, Colloid Interface Sci. 270 (2003)]. In Section 4, capillary imbibition of aqueous surfactant solutions into dry porous substrates is investigated from both theoretical and experimental points of view in the case of partial wetting [V. Straov, S. Zhdanov, M. Velarde, J. Colloid Interface Sci. 273 (2004) 589]. Cylindrical capillaries are used as a model of porous media for theoretical treatment of the problem. It is shown that if an averaged pore size of the porous medium is below a critical value, then the permeability of the porous medium is not influenced by the presence of surfactants at any concentration: the imbibition front moves exactly in the same way as in the case of the imbibition of the pure water. The critical radius is determined by the adsorption of the surfactant molecules on the inner surface of the pores. If an averaged pore size is bigger than the critical value, then the permeability increases with surfactant concentration. These theoretical conclusions are in agreement with experimental observations. In Section 5, the spreading of surfactant solutions over hydrophobic surfaces is considered from both theoretical and experimental points of view [V.M. Starov, S.R. Kosvintsev, M.G. Velarde, J. Colloid Interface Sci. 227 (2000) 185]. Water droplets do not wet a virgin solid hydrophobic substrate. It is shown that the transfer of surfactant molecules from the water droplet onto the hydrophobic surface changes the wetting characteristics in front of the drop on the three-phase contact line. The surfactant molecules increase the solid-vapor interfacial tension and hydrophilise the initially hydrophobic solid substrate just in front of the spreading drop. This process causes water drops to spread over time. The time of evolution of the spreading of a water droplet is predicted and compared with experimental observations. The assumption that surfactant transfer from the drop surface onto the solid hydrophobic substrate controls the rate of spreading is confirmed by experimental observations. In Section 6, the process of the spontaneous spreading of a droplet of a polar liquid over solid substrate is analyzed in the case when amphiphilic molecules (or their amphiphilic fragments) of the substrate surface layer are capable of overturning, resulting in a partial hydrophilisation of the surface [V.M. Starov, V.M. Rudoy, V.I. Ivanov, Colloid J. (Russian Academy of Sciences English Transaction) 61 (3) (1999) 374]. Such a situation may take place, for example, during contact of an aqueous droplet with the surface of a polymer whose macromolecules have hydrophilic side groups capable of rotating around the backbone and during the wetting of polymers containing surface-active additives or Langmuir-Blodgett films composed of amphiphilic molecules. It was shown that droplet spreading is possible only if the lateral interaction between neighbouring amphiphilic molecules (or groups) takes place. This interaction results in the tangential transfer of "the overturning state" to some distance in front of the advancing three-phase contact line making it partially hydrophilic. The quantitative theory describing the kinetics of droplet spreading is developed with allowance for this mechanism of self-organization of the surface layer of a substrate in the contact with a droplet.     

Sliding of liquid drops down an inclined solid surface

A liquid drop that partially wets a solid surface will slide down the plane when it is tilted beyond a critical inclination. Here we report the study of the sliding velocity of such a drop. Experiments for measuring the steady sliding velocity of different liquids of drops are performed. We then construct a scaling law that predicts the sliding velocity given the physical properties, wetting characteristics, and size of the drop. When the sliding velocity is low and the drop distortion due to inclination is small, the scaling law is shown to correctly model the functional dependency of the measured sliding velocity.     

Spreading of liquid drops over dry porous layers: complete wetting case

Spreading of small liquid drops over thin dry porous layers is investigated from both theoretical and experimental points of view. Drop motion over a porous layer is caused by an interplay of two processes: (a) the spreading of the drop over already saturated parts of the porous layer, which results in an expanding of the drop base; (b) the imbibition of the liquid from the drop into the porous substrate, which results in a shrinkage of the drop base and an expanding of the wetted region inside the porous layer. As a result of these two competing processes, the radius of the drop goes through a maximum value over time. A system of two differential equations is derived to describe the evolution with time of radii of both the drop base and the wetted region inside the porous layer. This system includes two parameters: one accounts for the effective lubrication coefficient of the liquid over the wetted porous substrate and the other is a combination of permeability and effective capillary pressure inside the porous layer. Two additional experiments are used for an independent determination of these two parameters. The system of differential equations does not include any fitting parameter after these two parameters are determined. Experiments were carried out on the spreading of silicone oil drops over various dry microfiltration membranes (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region inside the porous layer are monitored. All experimental data fell on two universal curves if appropriate scales are used with a plot of the dimensionless radii of the drop base and of the wetted region inside the porous layer on dimensionless time. The predicted theoretical relationships are two universal curves accounting quite satisfactorily for the experimental data. According to our theory prediction, (i) the dynamic contact angle dependence on the same dimensionless time as before should be a universal function and (ii) the dynamic contact angle should change rapidly over an initial short stage of spreading and should remain a constant value over the duration of the rest of the spreading process. The constancy of the contact angle on this stage has nothing to do with hysteresis of the contact angle: there is no hysteresis in our system. These conclusions again are in good agreement with our experimental observations.     

The possibility of different time scales in the dynamics of pore imbibition

To model the imbibition of liquids into porous solids, use is often made of the Lucas-Washburn equation, which relates the distance of penetration of a liquid at a given time to the pore radius, the viscosity and surface tension of the liquid, and the effective contact angle between the liquid and the solid. In this paper, we extend previous large-scale molecular dynamics simulations to show how this tool can be used to study the details of liquid imbibition, including the impact of the contact angle on the dynamics of penetration and the evolution of the internal flow field. In particular, we show that the asymptotic behavior of the contact angle versus time for a completely wetting liquid is given by approximately t(-1/4).     

Experimental investigation of the spontaneous wetting of polymers and polymer blends

The dynamics of polymeric liquids and mixtures spreading on a solid surface have been investigated on completely wetting and partially wetting surfaces. Drops were formed by pushing the test liquid through a hole in the underside of the substrate, and the drop profiles were monitored as the liquid wet the surface. Silicon surfaces coated with diphenyldichlorosilane (DPDCS) and octadecyltrichlorosilane (OTS) were used as wetting and partial wetting surfaces, respectively, for the fluids we investigated. The response under complete and partial wetting conditions for a series of polypropylene glycols (PPG) with different molecular weights and the same surface tension could be collapsed onto a single curve when scaling time based on the fluid viscosity, the liquid-vapor surface tension, and the radius of a spherical drop with equivalent volume. A poly(ethylene glycol) (PEG300) and a series of poly(ethylene oxide-rand-propylene oxide) copolymers did not show the same viscosity scaling when spread on the partially wetting surface. A combined model incorporating hydrodynamic and molecular-kinetic wetting models adequately described the complete wetting results. The assumptions in the hydrodynamic model, however, were not valid under the partial wetting conditions in our work, and the molecular-kinetic model was chosen to describe our results. The friction coefficient used in the molecular-kinetic model exhibited a nonlinear dependence with viscosity for the copolymers, indicating a more complex relationship between the friction coefficient and the fluid viscosity.     

Spreading of non-Newtonian liquids over solid substrates

The spreading of drops of a non-Newtonian liquid (Ostwald-de Waele liquid) over horizontal solid substrates is theoretically investigated in the case of complete wetting and small dynamic contact angles. Both gravitational and capillary regimes of spreading are considered. The evolution equation deduced for the shape of the spreading drops has self-similar solutions, which allows obtaining spreading laws for both gravitational and capillary regimes of spreading. In the gravitational regime case of spreading the profile of the spreading drop is provided.     

Wetting of TiC by Al-Cu alloys and interfacial characterization

The wetting behavior and the interfacial reactions that occurred between molten Al-Cu alloys (1, 4, 8, 20, 33, and 100 wt% Cu) and solid TiC substrates were studied by the sessile drop technique in the temperature range of 800-1130 degrees C. The effect of wetting behavior on the interfacial reaction layer was studied. All the Al-Cu alloys react with TiC at the interface forming an extensive reaction layer. The interface thickness varied with the samples, and depends on the temperature, chemical composition of the alloy and the time of the test. Wetting increases with increasing concentration of copper in the Al-Cu alloy at 800 and 900 degrees C. In contrast, at higher temperature such as 1000 degrees C wetting decreases with increasing copper content. The spreading kinetics and the work of adhesion were evaluated. The high values of activation energies indicated that spreading is not a simple viscosity controlled phenomenon but is a chemical reaction process. The spreading of the aluminum drop is observed to occur according to the formation of Al4C3, CuAl2O4, CuAl2, TiCux mainly, leading to a decreases in the contact angle. As the contact angle decreases the work of adhesion increases with increasing temperature. Al-Cu/TiC assemblies showed cohesive fracture corresponding to a strong interface. However, using pure Cu the adhesion work is poor, and the percentage of cohesion work is also too low (27-34%).     

Spreading of individual toner particles studied using in situ optical microscopy

This study develops and tests an experimental method to monitor in situ the dynamic spreading of individual toner particles on model substrates during heating, to simulate on laboratory scale the fusing sub-processes occurring in electrophotographic printing of paper. Real toner particles of cyan, magenta, yellow and black are transformed to perfect spheres by a temperature pre-treatment, then applied to the substrate, either high-energy clean glass or low-energy hydrophobised glass, and heated at rates up to 50 degrees C/min. The subsequent spreading as a function of time (and temperature) is recorded by an optical microscope and CCD camera mounted above the substrate, with the measured drop covering area used to calculate the corresponding toner-substrate-air contact angle. On the hydrophobic substrate the spreading is limited and equal for all four colours, while the substantially greater spreading on the hydrophilic substrate is accompanied by significant differences between the toner colours. In particular, the cyan and black toners are found to spread to almost twice the extent of the yellow particles. The dynamic spreading behaviour is interpreted in terms of complementary measurements of substrate and toner surface energy components and bulk toner rheology, and a simple empirical relation is proposed that fits very well the measurements for all toner and substrate types tested. In particular, the spreading relation is found to be determined only by the toner surface energy and its equilibrium contact angle, with no explicit dependence on toner viscosity.     

The effect of the temperature dependence of mobile phase viscosity on the retention time in series-coupled gas chromatographic columns

A previously established model of a system of series-coupled columns, incorporating the effects of temperature and mobile phase compressibility, is generalized to take the temperature dependence of the mobile phase viscosity coefficient into account. Expressions are reported for the linear flow velocity, the effective partition coefficient, and the retention time for both the case of constant mass flow rate and that of constant overall pressure drop. Numerical results indicate that the temperature dependence of the viscosity coefficient affects the retention time and pressure drop while the effective mass distribution coefficient remains essentially unchanged.     

Liquid–paper interactions during liquid drop impact and recoil on paper surfaces

The spreading and recoiling of water drops on several flat and macroscopically smooth model surfaces and on sized paper surfaces were studied over a range of drop impaction velocities using a high-speed CCD camera. The water drop spreading and recoiling results on several model hydrophobic and hydrophilic surfaces were found to be in agreement with observations reported in the literature. The maximum drop spreading diameter for those model surfaces at impact was found to be dependent upon the initial drop kinetic energy and the degree of hydrophobicity/hydrophilicity of the surface. The extent of the maximum drop recoiling was found to be much weaker for hydrophilic substrates than for hydrophobic substrates. Sized papers, however, showed an interesting switch of behaviour in the process of water drop impaction. They behave like a hydrophobic substrate when a water drop impacts on it, but like a hydrophilic substrate when water drop recoils. Although the contact angle between water and hydrophilic or hydrophobic non-porous surfaces changes from advancing to receding as reported in literature, the change of contact angle during water impact on paper surface is unique in that the level of sizing was found to have a smaller than expected influence on the degree of recoil. Atomic force microscopy (AFM) was used to probe fibres on a sized filter paper surface under water. The AFM data showed that water interacted strongly with the fibre even though the paper was heavily sized. Implications of this phenomenon were discussed in the context of inkjet print quality and of the surface conditions of sized papers. Results of this study are very useful in the understanding of inkjet ink droplet impaction on paper surfaces which sets the initial condition for ink penetration into paper after impaction.     

Experimental investigation of contact angle, curvature, and contact line motion in dropwise condensation and evaporation

Image-analyzing interferometry is used to measure the apparent contact angle and the curvature of a drop and a meniscus during condensation and evaporation processes in a constrained vapor bubble (CVB) cell. The apparent contact angle is found to be a function of the interfacial mass flux. The interfacial velocity for the drop during condensation and evaporation is a function of the apparent contact angle and the rate of change of radius of curvature. The dependence of velocity on the apparent contact angle is consistent with Tanner's scaling equation. The results support the hypothesis that evaporation/condensation is an important factor in contact line motion. The main purpose of this article is to present the experimental technique and the data. The equilibrium contact angle for the drop is found experimentally to be higher than that for the corner meniscus. The contact angle is a function of the stress field in the fluid. The equilibrium contact angle is related to the thickness of the thin adsorbed film in the microscopic region and depends on the characteristics of the microscopic region. The excess interfacial free energy and temperature jump were used to calculate the equilibrium thickness of the thin adsorbed film in the microscopic region.     

Anomalous viscoelasticity near the isotropic-nematic phase transition in liquid crystals

Recent optical Kerr effect experiments have shown that orientational relaxation of nematogens shows a pronounced slow down of the response function at intermediate times and also a power law decay near the isotropic-nematic (I-N) transition. In many aspects, this behavior appears to be rather similar to the ones observed in the supercooled liquid near-glass transition. We have performed molecular dynamics simulations of model nematogens (Gay-Berne with aspect ratio 3) to explore the viscoelasticity near the I-N transition and also investigated the correlation of viscoelasticity (if any) with orientational relaxation. It is found that although the viscosity indeed undergoes a somewhat sharper than normal change near the I-N transition, it is not characterized by any divergence-like behavior (like the ones observed in the supercooled liquid). The rotational friction, on the other hand, shows a much sharper rise as the I-N transition is approached. Interestingly, the probability distribution of the amplitude of the three components of the stress tensor shows anisotropy near the I-N transition-similar anisotropy has also been seen in the deeply supercooled liquid. Frequency dependence of viscosity shows several unusual behaviors: (a) There is a weak, power law dependence on frequency [eta(')(omega) approximately omega(-alpha)] at low frequencies and (b) there is a rapid increase in the sharp peak observed in eta(')(omega) in the intermediate frequency on approach to the I-N transition density. These features can be explained from the stress-stress time correlation function. The angular velocity correlation function also exhibits a power law decay in time. The reason for this is discussed.     

Influence of the Meniscus at the Bottom of the Solid Plate on Imbibition Experiments

The analysis of the experiments on liquid imbibition into porous solids carried out by using the increase of weight technique proves the influence that meniscus formation has on the experimental increase of weight when the bottom of the porous layer is put in contact with the free surface of the liquid used to perform the penetration experiments. This process (meniscus onset) has a temporary dependence due to the change of the distance between the inferior base of the plate and the free surface of the liquid as a result of the imbibition. We have also found the proper experimental conditions under which such temporary dependence can be minimized after a short time after the start of contact between the porous layer and the free surface of the liquid utilized in the penetration procedure. Thus, the weight increase because of the meniscus formation can be assumed as a constant during the greatest part of the experiments, allowing the use of the proper mathematical form of Washburn's equation in order to describe the imbibition of the liquid into the pores of the solid. Copyright 2001 Academic Press.     

Spreading and recoil of a surfactant-containing water drop on glass-supported alcohol films

In this paper we report the experimental observation of spreading and recoil of surfactant-containing water drops on various alcohol films supported on glass slides. The time evolution of spreading and recoil behavior was recorded by placing a web camera above the drop. We observed that the drop spread the fastest on CH3OH, followed by C2H5OH, and the slowest on i-PrOH. On the other hand, the recoil behavior was just the opposite. The drop recoiled the slowest on CH3OH and fastest on i-PrOH, while it recoiled in an intermediate time on C2H5OH. In addition, concentration of surfactant in the drop played a prominent role in the spreading and recoil time of the drop, with higher surfactant concentration making the drop spread and recoil faster. The time evolution of spreading velocity of the drop on different alcohol films at various surfactant concentrations occurred with a Gaussian distribution and the peak velocity was reached earliest on CH3OH followed by C2H5OH, while on i-PrOH it took the longest time. The recoil behavior was similar. The variation of velocity as a function of radius exhibited oscillatory behavior, indicating the existence of an interfacial phenomenon. We also report the observation that spreading of the drop occurred without observable fingering instability. Further, we observed by Fourier transform infrared (FTIR) spectroscopic measurements that the drop had mixed with the alcohol films as it spread. Miscibility of the alcohol in the film with the drop, alcohol evaporation cooling-induced temperature gradient, and Marangoni effect probably play important roles in the spreading and recoil behavior of the drop.     

Spreading Dynamics of Polydimethylsiloxane Drops: Crossover from Laplace to Van der Waals Spreading

The spreading dynamics of small polydimethylsiloxane (PDMS) drops was studied on substrates with varying surface energies. For experimental parameters near the wetting transition, we observed small PDMS drops of different drop volumes as a function of time using interference video microscopy. While for large drops the contact angle θ decreases with the well-established power-law relation θ approximately t(-0.3) (Tanner's law), the effect of dispersive van der Waals (VW) interactions must be taken into account when interpreting the evolution of small drops. Two signatures of the VW forces are observed. For a positive Hamaker constant, the disjoining pressure acts as an additional driving force, leading to an acceleration of droplet spreading as soon as the drop height becomes comparable to the range of the VW interactions. In addition, a precursor film forms ahead of the contact line, leading to an apparent volume loss, particularly noticeable for very small drops. Contact line pinning may be a problem and we describe its effect on our experimental results. We present a theory that discusses the interplay of surface tension and VW forces in the case of a spreading drop. This model predicts a new spreading regime for very thin drops, in agreement with our experimental results. Copyright 2001 Academic Press.     

Interfacial effects in the spreading kinetics of liquid droplets on solid substrates

Several theories deal with the spreading kinetics of liquids on solid substrate, most of which relate the rate of spreading to the surface tension and the viscosity of the liquid. Measurements of the spreading of a number of liquids exhibiting a wide range of surface tension and viscosity on dry soda-lime glass have been carried out to validate the proposed models. The measurements used a small droplet of constant volume to minimize gravitational effects. The contact radius was acquired as a function of time by an image analysis system. It was noted that power law theories describe the spreading rate for silicone oil on glass. However, significant departures were noted in the case of other liquids. Mechanistic considerations of our data suggest that equal volume droplets of similar surface tension and of diverse viscosity spread to the same area but at different rates. On the other hand, the spreading rate of glycerine, which exhibits incomplete spreading on glass, and that of silicone oil, with comparable viscosity behave similarly. These observations seemingly support the view that surface tension acts to retain the spherical shape of the droplet, whereas the difference between the solid-liquid and solid-vapor interfacial energies acts to enlarge the contact area. In the meantime, viscous dissipation acts to retard the spreading rate, past a constant rate regime.