Fundamental Relation for the Ideal Gas in the Gravitational Field and Heat Flow.

We formulate the first law of global thermodynamics for stationary states of the ideal gas in the gravitational field subjected to heat flow. We map the non-uniform system (described by profiles of the density and temperature) onto the uniform one and show that the total internal energy U(S*,V,N,L,M*) is the function of the following parameters of state: the non-equilibrium entropy S*, volume V, number of particles, N, height of the column L along the gravitational force, and renormalized mass of a particle M*. Each parameter corresponds to a different way of energy exchange with the environment. The parameter M* changes internal energy due to the shift of the centre of mass induced by the heat flux. We give analytical expressions for the non-equilibrium entropy S* and effective mass M*. When the heat flow goes to zero, S* approaches equilibrium entropy. Additionally, when the gravitational field vanishes, our fundamental relation reduces to the fundamental relation at equilibrium.

Parameters of State in the Global Thermodynamics of Binary Ideal Gas Mixtures in a Stationary Heat Flow.

In this paper, we formulate the first law of global thermodynamics for stationary states of the binary ideal gas mixture subjected to heat flow. We map the non-uniform system onto the uniform one and show that the internal energy U(S*,V,N1,N2,f1*,f2*) is the function of the following parameters of state: a non-equilibrium entropy S*, volume V, number of particles of the first component, N1, number of particles of the second component N2 and the renormalized degrees of freedom. The parameters f1*,f2*, N1,N2 satisfy the relation (N1/(N1+N2))f1*/f1+(N2/(N1+N2))f2*/f2=1 (f1 and f2 are the degrees of freedom for each component respectively). Thus, only 5 parameters of state describe the non-equilibrium state of the binary mixture in the heat flow. We calculate the non-equilibrium entropy S* and new thermodynamic parameters of state f1*,f2* explicitly. The latter are responsible for heat generation due to the concentration gradients. The theory reduces to equilibrium thermodynamics, when the heat flux goes to zero. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties.

Steady-state thermodynamics of a system with heat and mass flow coupling.

Equilibrium thermodynamics describes the energy exchange of a body with its environment. Here, we describe the global energy exchange of an ideal gas in the Coutte flow in a thermodynamic-like manner. We derive a fundamental relation between internal energy as a function of parameters of state. We analyze a non-equilibrium transition in the system and postulate the extremum principle, which determines stable steady states in the system. The steady-state thermodynamic framework resembles equilibrium thermodynamics.

Fundamental Relation for Gas of Interacting Particles in a Heat Flow.

There is a long-standing question of whether it is possible to extend the formalism of equilibrium thermodynamics to the case of nonequilibrium systems in steady-states. We have made such an extension for an ideal gas in a heat flow. Here, we investigated whether such a description exists for the system with interactions: the van der Waals gas in a heat flow. We introduced a steady-state fundamental relation and the parameters of state, each associated with a single way of changing energy. The first law of nonequilibrium thermodynamics follows from these parameters. The internal energy U for the nonequilibrium states has the same form as in equilibrium thermodynamics. For the van der Waals gas, U(S*,V,N,a*,b*) is a function of only five parameters of state (irrespective of the number of parameters characterizing the boundary conditions): the effective entropy S*, volume V, number of particles N, and rescaled van der Waals parameters a*, b*. The state parameters, a*, b*, together with S*, determine the net heat exchange with the environment. The net heat differential does not have an integrating factor. As in equilibrium thermodynamics, the steady-state fundamental equation also leads to the thermodynamic Maxwell relations for measurable steady-state properties.

Thermodynamics of stationary states of the ideal gas in a heat flow.

There is a long-standing question as to whether and to what extent it is possible to describe nonequilibrium systems in stationary states in terms of global thermodynamic functions. The positive answers have been obtained only for isothermal systems or systems with small temperature differences. We formulate thermodynamics of the stationary states of the ideal gas subjected to heat flow in the form of the zeroth, first, and second law. Surprisingly, the formal structure of steady state thermodynamics is the same as in equilibrium thermodynamics. We rigorously show that U satisfies the following equation dU = T*dS* - pdV for a constant number of particles, irrespective of the shape of the container, boundary conditions, the size of the system, or the mode of heat transfer into the system. We calculate S* and T* explicitly. The theory selects stable nonequilibrium steady states in a multistable system of ideal gas subjected to volumetric heating. It reduces to equilibrium thermodynamics when heat flux goes to zero.

You will know by its tail: a method for quantification of heterogeneity of bacterial populations using single-cell MIC profiling.

Severe non-healing infections are often caused by multiple pathogens or by genetic variants of the same pathogen exhibiting different levels of antibiotic resistance. For example, polymicrobial diabetic foot infections double the risk of amputation compared to monomicrobial infections. Although these infections lead to increased morbidity and mortality, standard antimicrobial susceptibility methods are designed for homogenous samples and are impaired in quantifying heteroresistance. Here, we propose a droplet-based label-free method for quantifying the antibiotic response of the entire population at the single-cell level. We used Pseudomonas aeruginosa and Staphylococcus aureus samples to confirm that the shape of the profile informs about the coexistence of diverse bacterial subpopulations, their sizes, and antibiotic heteroresistance. These profiles could therefore indicate the outcome of antibiotic treatment in terms of the size of remaining subpopulations. Moreover, we studied phenotypic variants of a S. aureus strain to confirm that the profile can be used to identify tolerant subpopulations, such as small colony variants, associated with increased risks for the development of persisting infections. Therefore, the profile is a versatile instrument for quantifying the size of each bacterial subpopulation within a specimen as well as their individual and joined heteroresistance.

Transient dynamics in the outflow of energy from a system in a nonequilibrium stationary state.

We investigate the thermal relaxation of an ideal gas from a nonequilibrium stationary state. The gas is enclosed between two walls, which initially have different temperatures. After making one of the walls adiabatic, the system returns to equilibrium. We notice two distinct modes of heat transport and associated timescales: one connected with a traveling heat front and the other with internal energy diffusion. At the heat front, which moves at the speed of sound, pressure, temperature, and density change abruptly, leaving lower values behind. This is unlike a shock wave, a sound wave, or a thermal wave. The front moves multiple times between the walls and is the dominant heat transport mode until surpassed by diffusion. We found that it can constitute an order 1 factor in shaping the dynamics of the outflow of internal energy. We found that cooling such a system is quicker than heating, and that hotter bodies cool down quicker than colder ones. The latter is known as the Mpemba effect.

Internal energy in compressible Poiseuille flow.

We analyze a compressible Poiseuille flow of ideal gas in a plane channel. We provide the form of internal energy U for a nonequilibrium stationary state that includes viscous dissipation and pressure work. We demonstrate that U depends strongly on the ratio Δp/p_{0}, where Δp is the pressure difference between inlet and outlet and p_{0} is the outlet's pressure. In addition, U depends on two other variables: the channel aspect ratio and the parameter equivalent to Reynolds number. The stored internal energy, ΔU=U-U_{0}, is small compared to the internal energy U_{0} of the equilibrium state for a moderate range of values of Δp/p_{0}. However, ΔU can become large for big Δp or close to vacuum conditions at the outlet (p_{0}≈0Pa).

Continuous nonequilibrium transition driven by heat flow.

We discovered an out-of-equilibrium transition in the ideal gas between two walls, divided by an inner, adiabatic, movable wall. The system is driven out-of-equilibrium by supplying energy directly into the volume of the gas. At critical heat flux we have found a continuous transition to the state with a low-density, hot gas on one side of the movable wall and a dense, cold gas on the other side. Molecular dynamic simulations of the soft-sphere fluid confirm the existence of the transition in the interacting system. We introduce a stationary state Helmholtz-like function whose minimum determines the stable positions of the internal wall. This transition can be used as a paradigm of transitions in stationary states and the Helmholtz-like function as a paradigm of the thermodynamic description of these states.

High-Throughput Monitoring of Bacterial Cell Density in Nanoliter Droplets: Label-Free Detection of Unmodified Gram-Positive and Gram-Negative Bacteria.

Droplet microfluidics disrupted analytical biology with the introduction of digital polymerase chain reaction and single-cell sequencing. The same technology may also bring important innovation in the analysis of bacteria, including antibiotic susceptibility testing at the single-cell level. Still, despite promising demonstrations, the lack of a high-throughput label-free method of detecting bacteria in nanoliter droplets prohibits analysis of the most interesting strains and widespread use of droplet technologies in analytical microbiology. We use a sensitive and fast measurement of scattered light from nanoliter droplets to demonstrate reliable detection of the proliferation of encapsulated bacteria. We verify the sensitivity of the method by simultaneous readout of fluorescent signals from bacteria expressing fluorescent proteins and demonstrate label-free readout on unlabeled Gram-negative and Gram-positive species. Our approach requires neither genetic modification of the cells nor the addition of chemical markers of metabolism. It is compatible with a wide range of bacterial species of clinical, research, and industrial interest, opening the microfluidic droplet technologies for adaptation in these fields.

Droplet-based digital antibiotic susceptibility screen reveals single-cell clonal heteroresistance in an isogenic bacterial population.

Since antibiotic resistance is a major threat to global health, recent observations that the traditional test of minimum inhibitory concentration (MIC) is not informative enough to guide effective antibiotic treatment are alarming. Bacterial heteroresistance, in which seemingly susceptible isogenic bacterial populations contain resistant sub-populations, underlies much of this challenge. To close this gap, here we developed a droplet-based digital MIC screen that constitutes a practical analytical platform for quantifying the single-cell distribution of phenotypic responses to antibiotics, as well as for measuring inoculum effect with high accuracy. We found that antibiotic efficacy is determined by the amount of antibiotic used per bacterial colony forming unit (CFU), not by the absolute antibiotic concentration, as shown by the treatment of beta-lactamase-carrying Escherichia coli with cefotaxime. We also noted that cells exhibited a pronounced clustering phenotype when exposed to near-inhibitory amounts of cefotaxime. Overall, our method facilitates research into the interplay between heteroresistance and antibiotic efficacy, as well as research into the origin and stimulation of heterogeneity by exposure to antibiotics. Due to the absolute bacteria quantification in this digital assay, our method provides a platform for developing reference MIC assays that are robust against inoculum-density variations.

Split or slip - passive generation of monodisperse double emulsions with cores of varying viscosity in microfluidic tandem step emulsification system.

We investigate the role of viscosities on the formation of double emulsion in a microfluidic step emulsification system. Aqueous droplets of various viscosities and sizes were engulfed in fluorocarbon oil and subsequently transformed into double droplets in the microfluidic step emulsifying device. We identify two distinct regimes of double droplet formation: (i) core droplets split into multiple smaller droplets, or (ii) cores slip whole into the forming oil shell. We show that the viscosity ratio of the core and shell phases plays a crucial role in determining the mode of formation of the double emulsions. Finally, we demonstrate that high viscosity of the core droplet allows for generation of double emulsions with constant shell thickness for cores of various sizes.

Gravity-driven microfluidic assay for digital enumeration of bacteria and for antibiotic susceptibility testing.

The alarming dynamics of antibiotic-resistant infections calls for the development of rapid and point-of-care (POC) antibiotic susceptibility testing (AST) methods. Here, we demonstrated the first completely stand-alone microfluidic system that allowed the execution of digital enumeration of bacteria and digital antibiograms without any specialized microfluidic instrumentation. A four-chamber gravity-driven step emulsification device generated ∼2000 monodisperse 2 nanoliter droplets with a coefficient of variation of 8.9% of volumes for 95% of droplets within less than 10 minutes. The manual workload required for droplet generation was limited to the sample preparation, the deposition into the sample inlet of the chip and subsequent orientation of the chip vertically without an additional pumping system. The use of shallow chambers imposing a 2D droplet arrangement provided superior stability of the droplets against coalescence and minimized the leakage of the reporter viability dye between adjacent droplets during long-term culture. By using resazurin as an indicator of the growth of bacteria, we were also able to reduce the assay time to ∼5 hours compared to 20 hours using the standard culture-based test.

Diffusion and flow in complex liquids.

Thermal motion of particles and molecules in liquids underlies many chemical and biological processes. Liquids, especially in biology, are complex due to structure at multiple relevant length scales. While diffusion in homogeneous simple liquids is well understood through the Stokes-Einstein relation, this equation fails completely in describing diffusion in complex media. Modeling, understanding, engineering and controlling processes at the nanoscale, most importantly inside living cells, requires a theoretical framework for the description of viscous response to allow predictions of diffusion rates in complex fluids. Here we use a general framework with the viscosity η(k) described by a function of wave vector in reciprocal space. We introduce a formulation that allows one to relate the rotational and translational diffusion coefficients and determine the viscosity η(k) directly from experiments. We apply our theory to provide a database for rotational diffusion coefficients of proteins/protein complexes in the bacterium E. coli. We also provide a database for the diffusion coefficient of proteins sliding along major grooves of DNA in E. coli. These parameters allow predictions of rate constants for association of proteins. In addition to constituting a theoretical framework for description of diffusion of probes and viscosity in complex fluids, the formulation that we propose should decrease substantially the cost of numerical simulations of transport in complex media by replacing the simulation of individual crowding particles with a continuous medium characterized by a wave-length dependent viscosity η(k).

Grooved step emulsification systems optimize the throughput of passive generation of monodisperse emulsions.

Microfluidic step emulsification passively produces highly monodisperse droplets and can be easily parallelized for high throughput emulsion production. The two main techniques used for step emulsification are: i) edge-based droplet generation (EDGE), where droplets are formed in a single, very wide and shallow nozzle, and ii) microchannel emulsification (MCE), where droplets are formed in many separated narrow nozzles. These techniques differ in modes of droplet formation that influence the throughput and monodispersity of produced emulsions. Here we report a systematic study of novel grooved step emulsifying geometries, a hybrid of MCE and EDGE architectures. We introduce partitions of different heights to a wide (EDGE-like) slit to establish optimal geometries for high-throughput droplet production. We demonstrate that the volume and monodispersity of the produced emulsion can be tuned solely by changing the height of these partitions. We show that the spacing of the partitions influences the size of the produced droplets, but not the population monodispersity. We also determine the moment of transition between two distinct droplet generation modes as a function of the geometrical parameters of the nozzle. The optimized grooved geometry appears to combine the advantages of both MCE and EDGE, i.e. spatial localization of droplet forming units (DFUs), high-throughput formation of tightly monodisperse droplets from parallel DFUs, and low sensitivity to variation in the flow rate of the dispersed phase. As a proof-of-concept we show grooved devices that for a 260-fold increase of flow rate produce droplets with volume increased by just 75%, as compared to 91% increase in volume over a 180-fold increase of flow rate of the dispersed phase in MCE devices. We also present the optimum microfluidic device geometry that almost doubles the throughput of an MCE device in the generation of nanoliter droplets.

Generalization of Clausius-Mossotti approximation in application to short-time transport properties of suspensions.

In 1983, Felderhof, Ford, and Cohen gave microscopic explanation of the famous Clausius-Mossotti formula for the dielectric constant of nonpolar dielectric. They based their considerations on the cluster expansion of the dielectric constant, which relates this macroscopic property with the microscopic characteristics of the system. In this article, we analyze the cluster expansion of Felderhof, Ford, and Cohen by performing its resummation (renormalization). Our analysis leads to the ring expansion for the macroscopic characteristic of the system, which is an expression alternative to the cluster expansion. Using similarity of structures of the cluster expansion and the ring expansion, we generalize (renormalize) the Clausius-Mossotti approximation. We apply our renormalized Clausius-Mossotti approximation to the case of the short-time transport properties of suspensions, calculating the effective viscosity and the hydrodynamic function with the translational self-diffusion and the collective diffusion coefficient. We perform calculations for monodisperse hard-sphere suspensions in equilibrium with volume fraction up to 45%. To assess the renormalized Clausius-Mossotti approximation, it is compared with numerical simulations and the Beenakker-Mazur method. The results of our renormalized Clausius-Mossotti approximation lead to comparable or much less error (with respect to the numerical simulations) than the Beenakker-Mazur method for the volume fractions below ϕ≈30% (apart from a small range of wave vectors in hydrodynamic function). For volume fractions above ϕ≈30%, the Beenakker-Mazur method gives in most cases lower error than the renormalized Clausius-Mossotti approximation.

Rotational self-diffusion in suspensions of charged particles: simulations and revised Beenakker-Mazur and pairwise additivity methods.

We present a comprehensive joint theory-simulation study of rotational self-diffusion in suspensions of charged particles whose interactions are modeled by the generic hard-sphere plus repulsive Yukawa (HSY) pair potential. Elaborate, high-precision simulation results for the short-time rotational self-diffusion coefficient, D(r), are discussed covering a broad range of fluid-phase state points in the HSY model phase diagram. The salient trends in the behavior of D(r) as a function of reduced potential strength and range, and particle concentration, are systematically explored and physically explained. The simulation results are further used to assess the performance of two semi-analytic theoretical methods for calculating D(r). The first theoretical method is a revised version of the classical Beenakker-Mazur method (BM) adapted to rotational diffusion which includes a highly improved treatment of the salient many-particle hydrodynamic interactions. The second method is an easy-to-implement pairwise additivity (PA) method in which the hydrodynamic interactions are treated on a full two-body level with lubrication corrections included. The static pair correlation functions required as the only input to both theoretical methods are calculated using the accurate Rogers-Young integral equation scheme. While the revised BM method reproduces the general trends of the simulation results, it significantly underestimates D(r). In contrast, the PA method agrees well with the simulation results for D(r) even for intermediately concentrated systems. A simple improvement of the PA method is presented which is applicable for large concentrations.

Transport properties of suspensions-critical assessment of Beenakker-Mazur method.

The Beenakker-Mazur method of calculation of transport coefficients for suspensions has been analyzed. The analysis relies on calculation of the hydrodynamic function and the effective viscosity with higher accuracy and comparison of these characteristics to the original Beennakker-Mazur results. Comparison to numerical simulations is also given. Our calculations go along with the idea of Beenakker and Mazur, but avoid unnecessary approximations. Our higher accuracy results differ significantly from results obtained initially by Beenakker and Mazur for volume fractions φ > 25%. Moreover, our results agree with the precise numerical simulations of Abade and Ladd for volume fractions φ < 15% and volume fractions φ ≈ 45%, whereas for volume fractions 15% < φ < 40%, we observe pronounced discrepancies.