Calculation Of Interfacial Tension Research Articles

Molecular Dynamics Calculation of Interfacial Tension in a Two-Phase Liquid Hydrocarbon–Water–Surfactant System: From Rarefied to Superdense Surfactant Monolayer

A method is proposed for calculating low interfacial tension (IFT) based on molecular dynamics simulation of systems with superdense packing of surfactant molecules at the water–liquid hydrocarbon interface. The interfacial tension was calculated by the molecular dynamics method using the all-atom and coarse-grained models in water–alkane (decane, dodecane) two-phase systems in the presence of various individual surfactants. The following ionic and nonionic surfactants were considered: sodium dodecyl sulfate (SDS), cetyltrimethylammonium chloride (CTAC), sodium dodecylbenzenesulfonate (SDBS), sodium decet-6 sulfate C10E6SO4Na, hexaethylene glycol monodecyl ether (C10E6), triethylene glycol monononadecyl ether (C19E3), and octapropoxypentaethylene glycol monododecyl ether (C12P8E5). It was shown that the interfacial tension decreases to zero when surfactant adsorption increases to the limiting values.

Calculation of IFT in porous media in the presence of different gas and normal alkanes using the modified EoS

Gas injection can increase oil recovery because the gas–oil interfacial tension is less than the water–oil interfacial tension (IFT) and tends to zero in the miscibility state. However, little information has been provided on the gas–oil movement and penetration mechanisms in the fracture system at the porosity scale. The IFT of oil and gas in the porous medium changes and can control oil recovery. In this study, the IFT and the minimum miscibility pressure (MMP) are calculated using the cubic Peng-Robinson equation of state that has been modified using the mean pore radius and capillary pressure. The calculated IFT and MMP change with the pore radius and capillary pressure. To investigate the effect of a porous medium on the IFT during the injection of CH4, CO2, and N2 in the presence of n-alkanes and for validation, measured experimental values in references have been used. According to the results of this paper, changes in IFT vary in terms of pressure in the presence of different gases and, the proposed model has good accuracy for measuring the IFT and the MMP during the injection of hydrocarbon gases and CO2. In addition, as the average radius of the pores gets smaller, the interfacial tension tends to lower values. This effect is different with increasing the mean size of interstice in two different intervals. In the first interval, i.e. the Rp from 10 to 5000 nm, the IFT changes from 3 to 10.78 mN/m and in the second interval, i.e. the Rp from 5000 nm to infinity, the IFT changes from 10.78 to 10.85 mN/m. In other words, increasing the diameter of the porous medium to a certain threshold (i.e. 5000 nm) increases the IFT. As a rule, changes in IFT affected by exposure to a porous medium affect the values of the MMP. In general, IFT decreases in very fine porous media, causing miscibility at lower pressures.

Fully biobased poly(lactic acid)/lignin composites compatibilized by epoxidized natural rubber

Biobased poly(lactic acid)/lignin (PLA/lignin) composites are limited by poor mechanical properties resulted from poor compatibility and low interfacial adhesion. Herein, we reported a novel approach to improve compatibility and interfacial adhesion of PLA/lignin composites via reactive compatibilization with epoxidized natural rubber (ENR) as a compatibilizer. Interfacial tension calculation indicated that lignin tended to act as interfacial phase between PLA and ENR, but morphology analysis demonstrated lignin was wrapped with a layer of ENR and dispersed in PLA matrix, which was attributed to the interfacial reaction of ENR with both PLA and lignin. The interfacial reaction was confirmed by Fourier transform infrared spectroscopy. The compatibility and interfacial adhesion between PLA and lignin were improved significantly by incorporation and increase in the content of ENR, as evidenced by the reduced interfacial gaps, blurry phase boundaries, and enhanced elastic response. As such, the mechanical properties of PLA/lignin composites were enhanced significantly. The tensile strength and elongation at break of PLA/lignin (W/W, 80/20) were improved by 15 % and 77 %, respectively, with the incorporation of only 1 wt% ENR. We believe this approach to compatibilize PLA/lignin composites is promising because it would not require costly modification of lignin and would not compromise the sustainability of composites.

Assessment of Inclusion Removal Ability in Refining Slags Containing Ce2O3

The elimination of inclusions in steelmaking processes has been widely studied. The removal of inclusions by slags containing the rare earth oxide Ce2O3 are studied using an integrated numerical model. The integrated model involves the inclusion motion model, interfacial tension calculation model, surface tension calculation model of slag, and the mass action concentration model, based on ion and molecule coexistence theory. The motion behaviors of both solid Al2O3 inclusions and 50%wtAl2O3–50%wtCaO liquid inclusions of varied sizes at CaO-Ce2O3-SiO2-Al2O3(-MgO) slag systems are evaluated. The results show that it is more difficult to remove the inclusions with smaller sizes and in slag with a higher viscosity. Liquid inclusions are more difficult to remove than solid inclusions. It is found that the CaO-Ce2O3-SiO2-Al2O3-MgO refining slag shows a better ability to remove Al2O3 inclusions than that of the CaO-SiO2-Al2O3-MgO slag. The reason for this is that the addition of the rare earth oxide Ce2O3 can decrease the viscosity of slags, as well as improving the wetting effects of slags on Al2O3 inclusions. For two slags systems, the CaO-Ce2O3-SiO2-Al2O3-MgO slag system shows a better ability to remove Al2O3 inclusions than the CaO-Ce2O3-SiO2-Al2O3 slag system. The addition of 5% to 8% Ce2O3 in a CaO-SiO2-Al2O3-MgO slag is an optimized case for industrial applications.

Hydrophobic and Hydrophilic Solid-Fluid Interaction

We propose a novel solid-fluid coupling method to capture the subtle hydrophobic and hydrophilic interactions between liquid, solid, and air at their multi-phase junctions. The key component of our approach is a Lagrangian model that tackles the coupling, evolution, and equilibrium of dynamic contact lines evolving on the interface between surface-tension fluid and deformable objects. This contact-line model captures an ensemble of small-scale geometric and physical processes, including dynamic waterfront tracking, local momentum transfer and force balance, and interfacial tension calculation. On top of this contact-line model, we further developed a mesh-based level set method to evolve the three-phase T-junction on a deformable solid surface. Our dynamic contact-line model, in conjunction with its monolithic coupling system, unifies the simulation of various hydrophobic and hydrophilic solid-fluid-interaction phenomena and enables a broad range of challenging small-scale elastocapillary phenomena that were previously difficult or impractical to solve, such as the elastocapillary origami and self-assembly, dynamic contact angles of drops, capillary adhesion, as well as wetting and splashing on vibrating surfaces.

Thermoplastic Elastomer based on Recycled HDPE/Ground Tire Rubber Interfacially Modified with an Elastomer: Effect of Mixing Sequence and Elastomer Type/Content

ABSTRACT The effect of interfacial interaction and processing methods (one-step and two-step) on the structure–property relationships of ternary blends was investigated. The addition of ethylene-vinyl acetate (EVA) and ethylene octene copolymer (POE) resulted in homogeneous structure and greatly improved rubber-like properties of recycled high-density polyethylene (rHDPE) and regenerated tire rubber (RR) blends. Interfacial tension calculations revealed a high tendency of POE to encapsulate rubber particles (complete wetting). The two-step processing contributed to perfect encapsulation of the rubber phase via a soft/thick POE interface resulting in uniform RR particle dispersion, high elongation at break (138.5%), and impact strength (394 J/m) of rHDPE/(RR/POE) (20/80) ternary blend.

Addition of surfactants to Low Salinity Waterflooding in microfluidics system to increase oil recovery

This study was conducted to investigate the phenomenon of oil removal from inside pores using a self-designed microfluidic test kit. An artificial micromodel chip as a representation of porous rocks has been created with a uniform pore structure design and made of PMMA (Polymethyl Methacrylate) material. The micromodel chip has a porosity of 27.8% as well as a permeability of 2.7 Darcy. By using the microfluidic test kit, this study has investigated how low salinity water (LSW) injection with MgCl2 divalent ions and the addition of anionic surfactant, linear alkylbenzene sulfonate mixed with nonionic surfactants, nonylphenol ethoxylate (NP-10) affects to oil recovery. The injection of LSW and surfactant solution was carried out with different injection stages, injection rates and surfactant solutions concentrations. Visual images during the injection process are recorded for analysis, which is the advantage of dynamic testing using this microfluidic test kit over conventional coreflooding. From this study, it is indicated that the selection of ions contained in LSW affects the success of LSW injection. Reducing the surfactant injection rate from 50 μL/min to 20 μL/min can increase the oil recovery from 1.27% to 4.29%. Oil recovery was also higher on surfactant injection which resulted in lower interfacial tension of the system based on the calculation of interfacial tension obtained from the Chun-Huh and Ghosh equations from the Winsor test. From all injection scenarios carried out in this study, the highest increase in oil recovery of 26.87% OOIP was obtained by injecting surfactant solutions directly in the secondary stage without prior LSW injection.

Melt behavior of polypropylene‐co‐ethylene composites filled with dual component of sago and kenaf natural filler

AbstractA dual component natural filler system featuring discrete function appears to be suitable for modulating smooth flow of pseudo‐plastic into formulated biodegradable polymer composite. Polypropylene‐co‐ethylene (PPcoE) filled with sago starch and different kenaf powder loadings were compounded using co‐rotating twin‐screw extruder to produce dual component polymer composite. The flow characteristics of the composites were measured using capillary rheometer and Bagley corrections were performed to achieve the true rheological data. In addition, data collected from the twin‐screw extruder were used to calculate the composite apparent rheological value. The flow properties of PPcoE filled with sago starch (PPcoE/sago) exhibited deviation from the common non‐Newtonian thermoplastic flow paths at high shear rates. Interestingly, the addition of kenaf powder into the PPcoE/sago starch blend stabilized the melted composite flow stream. The calculation of interfacial tension revealed that the flow corrections were related to the phase selective localization of kenaf particle, which preferably embedded between the sago particles rather than in the surrounding PPcoE phase thus reducing the effect of sago clustering. At higher extrusion temperature, the extruded composite indicated sign of flow instability due to factors of high amount of kenaf powder loading and the degradation of sago starch and PPcoE chains.

First-principles prediction of critical micellar concentrations for ionic and nonionic surfactants

The concentration of surfactant in solution for which micelles start to form, also known as critical micelle concentration is a key property in formulation design. The critical micelle concentration can be determined experimentally with a tensiometer by measuring the surface tension of a concentration series. In analogy with experiments, in-silico predictions can be achieved through interfacial tension calculations.We present a newly developed method, which employs first principles-based interfacial tension calculations rooted in COSMO-RS theory, for the prediction of the critical micelle concentration of a set of nonionic, cationic, anionic, and zwitterionic surfactants in water. Our approach consists of a combination of two prediction strategies for modelling two different phenomena involving the removal of the surfactant hydrophobic tail from contact with water. The two strategies are based on regular micelle formation and thermodynamic phase separation of the surfactant from water and both are required to take into account a wide range of polarity in the hydrophilic headgroup.Our method yields accurate predictions for the critical micellar concentration, within one log unit from experiments, for a wide range of surfactant types and introduces possibilities for first-principles based prediction of formulation properties for more complex compositions.

Calculation of interfacial tension of binary systems containing water and an organic component by group contribution methods

This work proposes a predictive method for estimating liquid-liquid interfacial tension (IFT) by group contribution methods of systems containing water and an organic component. The model for calculating the interfacial tension of liquid–liquid systems was derived using a general thermodynamic framework that was empirically extended by the introduction of an effective interfacial area. The effective interfacial area is defined for each component and takes into account the effects of groups of the other component present at the liquid-liquid interface. In addition, the interfacial molar volume was estimated by a mixing rule, using the molar volume of pure components in the liquid phase, weighted by the equilibrium compositions calculated by the UNIFAC model. The corresponding thermophysical properties were obtained by a group contribution method, in which a temperature-dependent binary parameter was proposed. This parameter is given by the sum of contributions related to each group present in the molecule, with values estimated from an experimental database of interfacial tension of water and different organic compounds. Therefore, the model can be used to predict the interfacial tension of binary mixtures in different temperatures through the knowledge of the structure of the organic component. The values of the model parameters were determined by a parameter estimation procedure based on an extensive database, covering 546 interfacial tension data of liquid-liquid binary systems containing water and an organic component. The model predicts interfacial tension data for binary systems containing organic compounds with different chemical functions in equilibrium with water, including oxygenated organic compounds, which are rarely predicted with good accuracy. The total mean absolute deviation obtained from this method was 1.20 mN⋅m−1, better than other predictive model and also better than some correlative models available in the literature.

Influence of different molecular weights of polyhexene‐1 on the morphology and rheology of cyclic olefin copolymer blends

AbstractThe morphology and rheological behavior of cyclic olefin copolymer (COC) blends with two molecular weights of polyhexene‐1 (PH‐1, PH‐1‐UH [ultra‐high]) were investigated at a wide range of compositions. Morphology of the blends at low concentrations of polyhexene‐1 s showed a droplet‐matrix structure and changed to a co‐continuous morphology at intermediate concentrations. The rheological Cole‐Cole plots and viscosity versus composition confirmed immiscibility of the blends. The interfacial interaction of blends phases was investigated and Complex viscosity and storage modulus versus frequency were measured and the results were consistent with high interfacial strength between the COC and high concentrations of PH‐1‐UH which increases the melt strength of the blends. Relaxation time spectra and Tan δ versus frequency curves were analyzed and these results were consistent with a high degree of entanglements between the COC and PH‐1‐UH chains in the PH‐1‐UH‐rich compounds which in turn increased the elasticity. Damping factor measurements and calculation of interfacial tension using emulsion models showed that in the COC‐rich blends, the interfacial interaction COC/PH‐1 blends is higher than that of the COC/PH‐1‐UH blends and thus the elasticity and particles form relaxation time of the PH‐1 blends are higher compared to PH‐1‐UH blends.

Fast Evaluation Model of Surfactant Flooding Based on Modification Fractional Flow Theory

At present, the numerical simulation of surfactant flooding is implemented by commercial software, which requires a lot of input data, resulting in slow calculation speed and heavy workload. So it was not suitable for surfactant flooding revovery evaluation in a large number of oilfields. This paper presents a fast evaluation model of surfactant flooding based on modification fractional flow theory. Viral expansion, interfacial tension calculation formula, and the relative permeability correction model are introduced to simulate the surfactant displacement mechanism. The simulation model is applied in a low permeability reservoir, and the simulation results are compared with the eclipse software results. Their results are similar, which show that this model presents certain reliability.

An alternative phase-field interfacial tension force representation for binary fluid systems

The Navier–Stokes/Cahn–Hilliard (NSCH) system of equations has been extensively used for investigating the dynamics of two-phase flows of Newtonian fluids. However, the accurate calculation of interfacial tension via NSCH has been perceptibly doubted, and thus, a successive solution of NSCH equations is rarely not accompanied by mesh adaptation techniques and complex numerical schemes. In this work, it is demonstrated that the cause of such a miscalculation of the interfacial tension is inherent when following the conventional way of coupling the Navier–Stokes with the Cahn–Hilliard equation in their dimensionless form, where the capillary number is defined by assuming that the fluid/fluid interface is flat and at equilibrium. Hence, an alternative NSCH model was developed for the more accurate computation of interfacial tension that does not rely on any such a priori assumptions, and it uses a more abstract coupling by accounting for the distribution of the binary system’s energy on the interfacial region. This model was implemented on two different cases: (i) an investigation of the effects of inertia and capillarity on the deformation of liquid drops in simple shear flow and (ii) a study of an interfacial instability due to viscosity stratification. To solve the set of governing equations, implicit time integration schemes based on finite differences were further developed and implemented. The results regarding the topological evolution of the fluid/fluid interface from both cases were additionally cross-validated with other methods from the literature as well as with the conventional NSCH model. The comparison suggests that our NSCH model indeed remedies the standard NSCH model, without the need of mesh adaptation or any complex numerical scheme, by more accurately computing the interfacial tension for binary systems consisting of incompressible, immiscible, and Newtonian fluids.

Probing the Interfacial Behavior of Type IIIa Binary Mixtures Along the Three-Phase Line Employing Molecular Thermodynamics.

Interfacial properties such as interfacial profiles, surface activity, wetting transitions, and interfacial tensions along the three-phase line are described for a Type IIIa binary mixture. The methodological approach combines the square gradient theory coupled to the statistical associating fluid theory for Mie potentials of variable range, and coarse-grained molecular dynamics simulations using the same underlying potential. The water + n-hexane mixture at three-phase equilibrium is chosen as a benchmark test case. The results show that the use of the same molecular representation for both the theory and the simulations provides a complementary picture of the aforementioned mixture, with an excellent agreement between the molecular models and the available experimental data. Interfacial tension calculations are extended to temperatures where experimental data are not available. From these extrapolations, it is possible to infer a first order wetting transition at 347.2 K, where hexane starts to completely wet the water/vapor interface. Similarly, the upper critical end point is estimated at 486.3 K. Both results show a very good agreement to the available experimental information. The concentration profiles confirm the wetting behavior of n-hexane along with a strong positive surface activity that increases with temperature, contrasting the weak positive surface activity of water that decreases with temperature.

Poly(methyl methacrylate)-induced Microstructure and Hydrolysis Behavior Changes of Poly(L-lactic acid)/Carbon Nanotubes Composites

Poly(L-lactic acid) (PLLA)-based composites exhibit wide applications in many fields. However, most of hydrophilic fillers usually accelerate the hydrolytic degradation of PLLA, which is unfavorable for the prolonging of the service life of the articles. In this work, a small quantity of poly(methyl methacrylate) (PMMA) (2 wt%–10 wt%) was incorporated into the PLLA/carbon nanotubes (CNTs) composites. The effects of PMMA content on the dispersion of CNTs as well as the microstructure and hydrolytic degradation behaviors of the composites were systematically investigated. The results showed that PMMA promoted the dispersion of CNTs in the composites. Amorphous PLLA was obtained in all the composites. Largely enhanced hydrolytic degradation resistance was achieved by incorporating PMMA, especially at relatively high PMMA content. Incorporating 10 wt% PMMA led to a dramatic decrease in the hydrolytic degradation rate from 0.19 %/h of the PLLA/CNT composite sample to 0.059 %/h of the PLLA/PMMA-10/CNT composite sample. The microstructure evolution of the composites was also detected, and the results showed that no crystallization occurred in the PLLA matrix. Further results based on the interfacial tension calculation showed that the enhanced hydrolytic degradation resistance of the PLLA matrix was mainly attributed to the relatively strong interfacial affinity between PMMA and CNTs, which prevented the occurrence of hydrolytic degradation at the interface between PLLA and CNTs. This work provides an alternative method for tailoring the hydrolytic degradation ability of the PLLA-based composites.

Gas Adsorption and Interfacial Tension with Classical Density Functional Theory

A classical density functional theory (DFT) that reduces to the perturbed chain statistical associating fluid theory equation of state has been implemented. This DFT implementation allows the use of molecular and binary interaction parameters fitted to the vapor–liquid equilibria of pure compounds and binary mixtures, respectively. Different functionals of dispersion forces are investigated and applied to the calculation of interfacial tension and adsorption of pure gases and their mixtures. The predictions of interfacial tension have been compared with both molecular simulations and experimental data, and good agreements have been seen. The DFT approaches also show good performance in correlating and predicting gas adsorption, and they are further compared with the multicomponent potential theory of adsorption and other DFT approaches available in the literature.

Confined fluid interfacial tension calculations and evaluations in nanopores

In this paper, confined fluid interfacial tensions (IFTs) in nanopores and their influential factors are studied. First, a new generalized equation of state (EOS) considering the pore radius, intermolecular interactions, and wall effect is developed analytically for calculating the thermodynamic phase behaviour of confined pure and mixing fluids in nanopores. Second, the modified model based on the new EOS and coupled with the parachor model, which also takes account for the capillary pressure and shifts of critical properties, is applied to calculate the IFTs in nanopores at different conditions. Third, the following four important factors are specifically studied to evaluate their effects on the IFTs in nanopores: feed gas to liquid ratio (FGLR), temperature, pore radius, and wall-effect distance. The newly-developed model is found to be accurate for vapour–liquid equilibrium (VLE) and IFT calculations in bulk phase and nanopores by comparing with the measured and calculated data in the literature. The IFTs in bulk phase of the pure and mixing hydrocarbon (HC) systems are always higher than those in nanopores. At low pressures, the calculated IFTs in nanopores from the new model are higher than those from the previously modified EOS, whereas they become almost equivalent at high pressures. The calculated IFTs of the simple HC systems in nanopores keep constant at different FGLRs while they are decreased by reducing the FGLRs for a multicomponent mixing HC system. Moreover, at low pressures, the gaseous CO2–mixing HC IFTs in bulk phase and nanopores are inferred to be lowered by increasing the temperature while the liquid/supercritical CO2–mixing HC IFTs may be increased. The temperature effect on the IFTs are weakened in nanopores at most pressures except for some extremely high pressure cases. The IFTs in nanopores are decreased with the reduction of pore radius but keep constant at δp/rp ≥ 1.0.

Measurement and modeling of isobaric vapor – Liquid equilibrium and isothermal interfacial tensions of ethanol + hexane + 2,5 – Dimethylfuran mixture

Isobaric vapor-liquid equilibrium data have been measured for the ternary system ethanol + hexane + 2,5 – dimethylfuran at 94 kPa and in the temperature range 330–344 K. Equilibrium determinations were performed in a vapor-liquid equilibrium still with circulation of both phases. The dependence of interfacial tensions of the ternary mixture on concentration was also experimentally determined at atmospheric pressure and 298.15 K, using the maximum differential bubble pressure technique.From the experimental results, it follows that the ternary mixture exhibits positive deviation from ideal behavior. Furthermore, the determined interfacial tensions exhibit negative deviations from the linear behavior, and ternary aneotropy is observed.The vapor-liquid equilibrium (VLE) data of the ternary mixture past the consistency test and were well correlated by Redlich–Kister expansion and predicted by the nonrandom two-liquid (NRTL), Wilson and universal quasichemical (UNIQUAC) activity coefficient models using binary parameters only. The interfacial tensions (IFT) were smoothed using the Myers-Scott expansion, showing an important contribution of parameters that describe three-body interactions, thus suggesting that the ternary interfacial tensions data cannot be initially predicted from the binary contributions.The experimental isobaric VLE and isothermal IFT data of the ternary mixture were accurately characterized by applying the linear square gradient theory to the Peng–Robinson Stryjek–Vera equation of state (EoS) appropriately extended to mixtures by means of the modified Huron–Vidal mixing rule. This model allows directly transferring the experimental excess Gibbs energy function to the EoS model for equilibrium and interfacial tension calculation purposes. This theoretical framework shows that both experimental VLE and IFT data can be accurately predicted by using binary contributions and provide a route to interpolate values phase equilibrium and interfacial tension.

A rapid method for interfacial tension calculation between rock plug and crude oil based on contact angles, application for EOR

ABSTRACTInterfacial tension and contact angle are two specific important parameters to take decisions for enhanced oil recovery, for instance, proper chemicals to use for surface tension reduction, expected wettability of solids, interaction between crude oil and rock. For this purpose, the article presents a method for easy calculation of the solid-liquid interfacial tension based on contact angle measurements applying Neumann's correlation and Young's equation. The main idea stands on the calculation of the rock parameters, like wettability, with known substances and extend these results to crude oils. It was possible, based on the results obtained, to establish a relationship between solid-liquid interfacial tension and contact angle for the crude oil – rock system, which can definitively be used for the calculation of interfacial tension of any other fluid spread out on the same kind of rock. A linear regression was obtained with an accuracy as good as R2 = 0.9989. Viscosity as a function of contact angle could also be obtained for the studied crude oils in the same kind of rock.

General approach for solving the density gradient theory in the interfacial tension calculations

Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory.