Renormalization of the Completely Asymmetric Fluctuation Equation (CAFE)

Abstract

Embodiment simulates liquid mixture. Embodiment begins by receiving a free energy curve of a liquid mixture comprising two components. Based on the received free energy curve, a partition function is constructed that describes the fluctuations of the mole fractions of the two components in the liquid mixture. Then, a renormalized free energy curve of the liquid mixture is calculated using the constructed partition function. The calculated renormalized free energy curve is then used (i.e., applied) to simulate the behavior of the liquid mixture. Computer automated simulation systems and apparatus are example applications of the methods and disclosed techniques of the present invention.

Description

[0001] Related Applications

[0002] This application claims the benefit of U.S. Provisional Application No. 63 / 110,431, filed on November 6, 2020. The entire teachings of the above application are incorporated herein by reference. Background Art

[0003] Simulation (and in particular computer-based fluid simulation) has become an important part of many industrial development processes. The functions provided by ) are used to develop products such as liquids in many industries (for example, the pharmaceutical and chemical product industries). In addition, the results of these liquid simulations are also used to modify or control product manufacturing processes and improve systems such as manufacturing efficiency.

[0004] While the use of computer-based liquid simulations has become widespread, such methods and systems may benefit from improvements in the accuracy of the simulations. Summary of the invention

[0005] The embodiments provide such improvements. In particular, the embodiments improve the quantitative simulation of liquid mutual solubility and solve problems in computer-based simulation of thermodynamic equilibrium of liquid systems. The embodiments accurately predict the mutual solubility of two liquids near the critical solution temperature.

[0006] One such embodiment, which may be referred to herein as CAFE (Complete Asymmetric Fluctuation Equation), constructs a partition function based on the total free energy curve g(x;T) of a liquid mixture of two components, which covers all possible symmetric and asymmetric linear fluctuations of the mole fractions around the average mole fraction X at temperature T. Such an embodiment converts the renormalized free energy curve g ren (x) is calculated as the constructed partition function multiplied by the negative logarithm of the thermal energy RT, where R is the universal gas constant. One exemplary embodiment utilizes a scaling parameter λ for temperature T, which is a universal constant or a universal linear function of the characteristics that can be obtained from the unrenormalized g(x;T) curve. Continuing with the description, standard tangent construction methods that implement basic thermodynamic calculations are then used from g ren (x; T) calculates the renormalized liquid-liquid equilibrium (LLE) curve. In turn, this renormalized LLE curve is used to simulate the liquid mixture.

[0007] Another example embodiment relates to a computer-implemented method for simulating a liquid mixture. The method begins by receiving a free energy curve for a liquid mixture comprising two components. Based on the received free energy curve, a partition function is constructed that describes the fluctuations in the mole fractions of the two components in the liquid mixture. In turn, a renormalized free energy curve for the liquid mixture is calculated using the constructed partition function. The calculated renormalized free energy curve is then used to simulate the behavior of the liquid mixture. This simulation application of the calculated or derived renormalized free energy curve provides advantages and efficiencies that have heretofore not been achieved in simulation technology.

[0008] One embodiment receives a free energy curve in a mathematical representation. In addition, because the method is computer-implemented, the free energy curve can be received from any point communicatively coupled to a computing device implementing the method. According to one embodiment, the constructed partition function covers symmetric and asymmetric fluctuations of the mole fractions of the two components around the average mole fraction. In one embodiment, the renormalized free energy curve is the constructed partition function multiplied by the negative logarithm of the molar thermal energy RT. In another example embodiment, the renormalized free energy curve is a function of a temperature scale parameter. In such an embodiment, the temperature scale parameter can be a constant or a function of a property determined based on the received free energy curve of the liquid mixture.

[0009] In one embodiment of the method, simulating the behavior of the liquid mixture using the calculated renormalized free energy curve includes predicting the liquid-liquid equilibrium of the liquid mixture using a tangent construction method. Simulations performed using the calculated renormalized free energy curve can be used to predict various different characteristics of the behavior of the liquid mixture. For example, embodiments can predict critical solution temperatures, such as lower critical solution temperatures and upper critical solution temperatures, and / or renormalized equilibrium compositions of two components, among other examples.

[0010] An example embodiment determines a more realistic shape of the predicted liquid-liquid equilibrium curve for a liquid mixture at different temperatures. This can be used in existing simulation applications and programs (e.g., ) to improve the results determined by these existing applications, such as liquid-liquid equilibrium curves. The embodiments can be used for liquid processing and screening simulations. The embodiments can be used to determine the best mixture and the conditions for creating the mixture. Such embodiments can be used in a variety of fields and applications, including chemical and pharmaceutical applications, as well as any application that utilizes liquid mixture simulations.

[0011] Another embodiment relates to a system comprising a processor and a memory having computer code instructions stored thereon. In such an embodiment, the processor and the memory are configured to cause the system to implement any embodiment or combination of embodiments described herein using the computer code instructions.

[0012] Yet another embodiment of the present invention relates to a cloud computing implementation for simulating liquid mixtures. Such an embodiment relates to a computer program product executed by a server communicating with one or more clients over a network, wherein the computer program product includes instructions that, when executed by one or more processors, cause the one or more processors to implement any of the embodiments described herein. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The foregoing will become apparent from the following more particular description of example embodiments, as illustrated in the accompanying drawings in which like reference numerals refer to the same parts throughout the different views. The drawings are not necessarily drawn to scale, emphasis instead being placed upon illustrating the embodiments.

[0014] Figure 1 is a plot showing the free energy profile of a liquid mixture at various temperatures that cannot be simulated using existing methods.

[0015] Figure 2 is a flow chart of a method for simulating a liquid mixture according to an embodiment.

[0016] Figure 3-5 are graphs showing simulation results determined using the embodiment and the prior art method.

[0017] Figures 6A-6C The surface polarity of compounds that can be simulated using the Examples is shown.

[0018] Figure 7 is to use existing simulation technology and embodiments of the present invention Figures 6A-6C Plot of the simulation results for the compound .

[0019] Figure 8 is a simplified block diagram of a computer system for simulating liquid mixtures according to an embodiment.

[0020] Fig. 9 is a simplified block diagram of a computer network environment in which embodiments of the present invention may be implemented. DETAILED DESCRIPTION

[0021] Following is a description of example embodiments.

[0022] As described above, embodiments of the present invention provide improved computer-based simulations of liquid mixtures. Liquid mixtures of compounds typically have: a temperature range in which the two compounds are miscible (forming a homogeneous solution) at any concentration; and other temperature regions in which the compounds decompose into two or more subvolumes (phases) with different compositions. The composition range between equilibrium concentrations is called the miscibility gap. The coexistence of multiple phases at one temperature is called liquid-liquid equilibrium (LLE). Typically, liquids are miscible above the upper critical solution temperature (UCST) and immiscible below the UCST. However, liquids may also be miscible below the lower critical solution temperature (LCST) and immiscible above the LCST. Here, the term critical solution temperature (CST) is used to refer to both the UCST and the LCST.

[0023] For liquid mixtures, as long as the temperature is far enough from the CST, the phases are well separated by the maxima in the total free energy profile (TFEP) of the mixture (i.e., the total free energy of the system as a function of the concentration variable), and the composition of each phase is fixed to the resulting minimum. However, if the temperature of the liquid mixture approaches the CST, the barriers between the minima become lower and lower, and the system (i.e., the liquid mixture) fluctuates over a wide range of composition space. This phenomenon, which also occurs in the gas-liquid transition of pure fluids, gives rise to long-range composition fluctuations, also called critical density fluctuations. These fluctuations lead to an effective smoothing of the free energy landscape and a reduction of the barriers, and, therefore, the emergence of critical fluctuations is a self-supporting process, which leads to a rapid disappearance of the LLE gap.

[0024] Existing analytical models for calculating the total free energy (TFE) of liquid mixtures do not include the effects of critical fluctuations and, therefore, produce UCSTs that are too high and LCSTs that are too low. In addition, existing methods overestimate the width of the miscibility gap at temperatures close to the CST. This limits the practical use of predictive models in the simulation, development, and optimization of dissolution and separation processes.

[0025] Figure 1 The plot 100 in FIG. 1 illustrates the critical fluctuations that cannot be simulated using existing methods. The plot 100 shows the nearest neighbor interaction energy G at four different temperatures (1.5 curve 101, 2 curve 102, 2.26 curve 103, 2.46 curve 104). AB The free energy curve (g(x) curve) of a simple cubic lattice fluid is represented by G AB Energy units quantified. Figure 1The g(x) curves 101, 102, 103 and 104 shown in are taken from COSMOSPACE and have the mathematical form:

[0026] and

[0027] in,

[0028] and

[0029] is the surface fraction, A1 and A2 are the molecular surface areas, and ω is given by:

[0030]

[0031] Among them, E 12 is the interaction energy between type 1 and type 2 segments.

[0032] Lines 105, 106, 107 and 108 in plot 100 represent energy levels T / T relative to the minimum crit Δ crit , where Δ crit is the barrier height at the critical temperature, which corresponds roughly to RT / 300.

[0033] At temperatures well below the critical temperature (curve 101), the system is fairly confined to regions close to the minimum of the total free energy (the minimum of curve 101), which are well separated by free energy barriers. As the temperature increases, the thermal energy increases, but more importantly the free energy barriers decrease, and the minima become increasingly shallow. In this way, more and more states in the composition space become thermally available, and the system fluctuates between these states. At a certain temperature (i.e., the true UCST), the barriers become so small that the fluctuations are no longer confined to the individual minimum regions, and thus it is impossible to distinguish separate regions of different composition, i.e., there are no separate phases. At this point, the system (liquid mixture) is macroscopically homogeneous, although there is a small free energy barrier separating the original phases. This barrier disappears only at the UCST of an analytical free energy model (in this case the COSMOSPACE model [6]). When performing simulations on such liquids, existing methods are unable to accurately simulate the fluctuations and therefore are generally unable to accurately predict the true UCST.

[0034] Several methods have been developed in an attempt to correctly determine the simulation, e.g. the correct UCST. The procedures and algorithms that convert the LLE curve from its erroneous analytical shape to its actual shape due to critical fluctuations are called renormalization methods. Existing renormalization methods require fitting many adjustable parameters to experimental data for the particular system (liquid mixture) under consideration. Therefore, existing renormalization methods cannot be used in a predictive manner, i.e., to predict CST and LLE curves for systems that have not been previously explored experimentally.

[0035] The problem of quantitative description of the mutual solubility of liquids close to their upper (or lower) critical solution temperature (UCST or LCST, respectively) is a long-standing problem known for more than a century [1-3]. All analytical free energy methods, which generally give a reasonable description of the solubility curves (as a function of temperature, also called LLE (liquid-liquid equilibrium) curves), fail strongly above this limit for temperatures below 90% of the UCST. The predicted LLE curves close too slowly (where the critical exponent is 0.5) typically resulting in predicted UCSTs that are about 10% too high, which can easily reach 30–50 K. At the same time, experimental LLE curves close very rapidly with a critical exponent of about 0.32. Such inaccurate predictions are often a barrier to the use of prediction methods for practical applications in chemical engineering (e.g., the development of chemical separation processes).

[0036] Many interpolation methods have been developed, based in part on complex theories such as the renormalization group theory by Ginzburg and Landau [4]. However, these existing methods require many experiments on the same system in order to fit numerous model parameters to the experimental data. Moreover, even with many parameters, it is not easy to adequately describe the anomalous behavior in the case of a critical exponent of 0.32. Most methods that accurately approximate the UCST fail in the crossover region with the analytically well-described region.

[0037] Molecular simulations in both variants (molecular dynamics simulations and Monte Carlo simulations) are able to describe critical behavior if sufficiently large systems are simulated for a long time. However, this requires very long simulation times. In addition, the thermodynamic accuracy of such force field-based simulation methods is usually much lower than the accuracy of analytical free energy models. Therefore, such simulations are not practical at all for industrial applications (i.e., simulation, development and manufacture of real-world liquid mixtures).

[0038] What is still missing is a prediction method that can be effectively applied to free energy curves generated by good predictive free energy models to provide robust, improved LLE predictions close to UCST without using any experimental data on the system.

[0039] Embodiments of the present invention address the above-mentioned shortcomings in the art and provide such functionality. Figure 2 2 is a flow chart of one such computer-implemented method 220 for simulating liquid mixtures. The method 220 begins at step 221: receiving a free energy curve for a liquid mixture comprising two components. Because the method 220 is computer-implemented, the free energy curve can be received at step 221 from any point communicatively coupled to a computing device implementing the method 220. The curve received at step 221 can relate to any binary liquid mixture known in the art, and likewise, the two components can be any components known in the art. For example, cyclooctane and pentafluorobutane, etc. In one embodiment of the method 220, the free energy curve received at step 221 is in the form of a mathematical representation. For example, the free energy curve (unrenormalized free energy curve) received at step 221 can be in the following form:

[0040] g(x1)=-kT(lnx1+lnγ1+lnx2+lnγ2)

[0041] Where k is the Boltzmann constant, T is the temperature, x1 and x2 are the mole fractions, and γ1 and γ2 are the activity coefficients of the two components. The latter can be taken from any analytical activity coefficient model, such as NRTL, UNIQUAC, Wilson and van Laar, etc., or from the COSMOSPACE model. The logarithmic activity coefficient can be taken from the NRTL model (Non-Random-Two-Liquid-Model, Renon H., Prausnitz JM: Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures, AIChE J., 14(1), S.135–144, 1968). If the logarithmic activity coefficient is taken from the NRTL model, the mathematical expression of the logarithmic activity coefficient is given by the following formula:

[0042]

[0043]

[0044] Among them, τ ij and G ij It is a parameter adjusted for the system.

[0045] return Figure 2In step 222, a partition function describing the fluctuations of the mole fractions of the two components in the liquid mixture is constructed based on the received free energy curve. In one embodiment of the method 220, the partition function is constructed in step 222 using the functions described below in relation to equations (2)-(7). According to one embodiment of the method 220, the partition function constructed in step 222 using equations (2)-(7) covers or takes into account the symmetric and asymmetric fluctuations of the mole fractions of the two components around the average mole fraction.

[0046] To proceed, in step 223, a renormalized free energy curve of the liquid mixture is calculated using the constructed partition function generated or output by step 222. An example of a renormalized free energy curve calculated in step 223 is given by equation (1) below. According to one embodiment, the renormalized free energy curve calculated in step 223 is the constructed partition function multiplied by the negative logarithm of the molar thermal energy RT. In yet another example embodiment, the renormalized free energy curve calculated in step 223 is a function of a temperature scale parameter. In such an embodiment of method 220, the temperature scale parameter can be a constant or a function of a characteristic determined based on the free energy curve of the liquid mixture received in step 221.

[0047] In step 224, the behavior of the liquid mixture is simulated using the calculated renormalized free energy curve from step 223. In an embodiment of method 220, simulating the behavior of the liquid mixture using the calculated renormalized free energy curve in step 224 includes: predicting the liquid-liquid equilibrium of the liquid mixture using a tangent construction method. The embodiment can use any implementation of the tangent construction method known in the art, such as the function described in [7] Lee, DD, JH Choy, and JK Lee. "Computer generation of binary and ternary phase diagrams via a convex hull method." Journal of phase equilibria 13.4 (1992): 365-372. Performing the simulation in step 224 can predict a variety of different characteristics of the behavior of the liquid mixture. For example, an embodiment of method 220 can predict the critical solution temperature, the lower critical solution temperature, the upper critical solution temperature and / or the renormalized equilibrium composition, that is, the mutual solubility of two components (compounds), etc.

[0048] In one embodiment of the invention, such as method 220, the renormalization of the free energy function g(x) is described by the following equation:

[0049]

[0050]

[0051] Equation 1 shows that the renormalized free energy of the system (including the free energy contribution caused by critical fluctuations) is calculated based on the logarithm of the partition function. According to basic statistical thermodynamics, this is even the definition of free energy. The field of statistical thermodynamics is to find the correct method for enumerating the states of the system.

[0052] Equation 2 shows that for a given total composition x, any fluctuation maintains that average concentration. Traditional renormalization theory only considers symmetric fluctuations consisting of two equal regions with compositions x + δ and x - δ. Since negative compositions are impossible, this definition restricts δ to the range 0 < δ < x, which means that no fluctuation can reach a composition greater than 2δ. Another physically unrealistic aspect of the symmetric (x + δ, x - δ) fluctuations that is usually not considered is the free energy in the cross - region between the extremes. However, fluctuations inevitably have such cross - regions. The embodiments overcome these problems for the first time in the art by using a two - dimensional representation of the fluctuations, which are characterized by their extreme compositions y < x and z > x, where is the average free energy of the considered fluctuation (based on the underlying analytical free energy model).

[0053] For a given set of mole fractions x, y, and z, the free energy One possible assumption for the shape of the fluctuation of the free energy is given by a linear weight function. The shape of the fluctuation of the free energy can also be given by an exponential weight function or a piecewise linear weight function, etc. In such an embodiment, the free energy is given by the integral of all intermediate mole fractions between y and z with a linear interpolation weight function:

[0054]

[0055] The norm of the weight function is unity (1), and the average composition is x, resulting in Equations 4 and 5:

[0056] where and (4) and (5)

[0057] Equations (4) and (5) can be solved analytically, resulting in Equations 6 and 7:

[0058] and (6) and (7)

[0059] Thus, these equations 1-7 describe a complete model for calculating the renormalized free energy. The equilibrium composition can be derived from this model by standard methods of thermodynamics, such as the search for two minima of the free energy for a symmetric system, or a tangent search for an asymmetric system. The above equations use the scaling factor λ as the only empirical parameter. 1 / λ can be interpreted as the number of molecules required to define the composition of a bulk region, since a single molecule does not define the composition.

[0060] For the following Figure 3-5 In the example described, λ = 80. This can be interpreted as approximately 80 molecules creating enough volume to define the composition in composition space. Figure 3-5 The results in compare the results determined using an embodiment of the present invention with the results determined using a lattice-Monte-Carlo (LMC) technique.

[0061] Figure 3 Plot 330 in shows the results of a simulation to determine the LLE curves of four Ising analogs of 3D lattice diamond, simple cubic, face-centered cubic (fcc), and body-centered cubic (bcc), with corresponding coordination numbers, i.e., the number of nearest neighbor molecules, of 4, 6, 8, and 12, respectively. Lines 331a-d are LMC results, which can be considered substantially accurate in these Ising cases. Lines 332a-d are the results of corresponding COSMOSPACE calculations. Lines 333a-d are the results generated using an embodiment of the present invention (e.g., CAFE, which may include method 220). Plot 330 shows that an embodiment of the present invention provides a good fit for all four cases.

[0062] Figure 4 Plot 440 in shows the LLE curve and the renormalized LLE curve (determined using the example) for a lattice fluid mixture of a polar pseudo-compound (pseudo-water) and a non-polar pseudo-compound (pseudo-methane). Figure 4 , the LMC results (lines 441 and 442) were determined using Hahn & Klamt [5]. The unrenormalized free energy curve 444 and the LLE curve 443 were generated using the COSMOSPACE method [6]. In plot 440, the LLE curves 441-446 become asymmetric due to different interactions. Lines 445 and 446 show the results generated using an embodiment of the present invention (i.e., CAFE). Although the results 445 and 446 generated using an embodiment of the present invention do not perfectly match the LMC results 441 and 442, the results 445 and 446 generated using the embodiment give a good quantification of the critical fluctuation effect. Note that these LMC results 441 and 442 also have significant uncertainty.

[0063] Figure 5 Plot 550 in Figure 550 illustrates the LLE curves for a size-asymmetric system, where compound 1 consists of 1 cube and compound 2 consists of 1, 2, 3, and 4 linearly connected cubes, respectively (from left LLE to right LLE). The LMC (series 551) calculations were performed by Max Hahn. Series 552 are COSMOSPACE results using The 553 series are CAFE results, i.e., results generated based on COSMOSPACE+COSMOcombi free energy curves using an embodiment of the present invention. Figure 5 It is shown that the CAFE results (553) are not completely consistent with the LMC (551), but in all cases are considerable improvements over the COSMOSPACE LLE curves (552).

[0064] One embodiment corrects (i.e., renormalizes) the predicted critical fluctuation miscibility gap (LLE point) of a liquid mixture based on a total free energy curve calculated using an accurately predicted paired surface segment fluid phase thermodynamic model that has not been fit to experimental data or Monte Carlo simulation data for the liquid mixture under consideration.

[0065] Embodiments can be used predictively without the need for system-specific adjustable parameters. Embodiments are computationally efficient and robust. Embodiments also provide renormalized total free energy curves, which have a variety of applications. For example, the renormalized total free energy curve can be used to calculate the renormalized activity coefficient, which can be used for LLE predictions in thermodynamic simulations, and other examples. To date, there is no method in the art to provide such a renormalized free energy curve.

[0066] Embodiments of the present invention (e.g., method 220) are computer-implemented. Thus, embodiments may be implemented using any combination of a processor and a computer memory (which is programmed in such a way as to perform the functions described herein). For example, an embodiment implementing the CAFE renormalization of the free energy function g(x) may be implemented by the following FORTRAN subroutine:

[0067] FORTRAN subroutine encoding the CAFE renormalization of the free energy function g(x)

[0068]

[0069]

[0070]

[0071] An example application of an embodiment of the present invention is provided below. In such an illustrative embodiment, a refrigerator manufacturer plans to optimize the refrigeration medium by using a mixture of refrigerants. The process needs to operate in the homogeneous mixture region, and therefore, it is critical for him to predict the miscibility range of the mixture. Such predictions can be made using the example embodiment. Note that the example implementation described below is only an example use of the embodiment, and the embodiment can be used for any application where a computer-based or computer-automated liquid simulation is desired.

[0072] This exemplary embodiment simulates a mixture of the alkane cyclooctane and the partially fluorinated compound 1,1,1,3,3-pentafluorobutane (PFB). In such an embodiment, existing simulation applications (e.g., in their commercially available The predictive thermodynamic model COSMO-RS in the embodiment is used to calculate the liquid-liquid equilibrium (LLE) curve of the binary mixture. The input requires DFT / COSMO surface polarity information of the two compounds, which is calculated using existing applications, such as the quantum chemistry program TURBOMOLE (or other equivalent quantum chemistry programs). The results of these calculations for the two compounds cyclooctane and PFB are shown in Figures 6A-6C middle. Fig. 6A The surface polarity 660a, ie the DFT / COSMO surface polarization charge density of cyclooctane, is illustrated. For PFB, two relevant intramolecular geometries, the so-called conformations, are considered. Figure 6B and 6C The surface polarities 660b and 660c of two relevant geometries of the PFB are depicted, respectively.

[0073] To continue with this illustrative example, the user can load these COSMO results into program (or other such program to calculate LLE) and select the LLE calculation panel. Then set the starting temperature for the LLE search (-100°C in the example), and The program automatically calculates the unrenormalized LLE points at predefined temperature steps. At each temperature, the program calculates the total free energy curve. The free energy curve can then be processed as described herein, for example, using method 220, to determine the renormalized LLE curve. In an example implementation, A local addition to the program (or other such program for determining the LLE) is configured to apply an embodiment of the present invention (eg, the CAFE renormalization equations 1-7 described above or a CAFE subroutine) and search for the LLE point using a tangent procedure.

[0074] Figure 7Plot 770 in shows an unrenormalized miscibility curve 771 with typical parabolic behavior near the UCST, which is about 295 K. Applying an embodiment of the present invention (e.g., CAFE reformation procedure) to the original free energy curve of produces an LLE curve 772 having a typical flat region near the UCST (which is now 275.6 K). Thus, by applying an example embodiment of the present invention, the correct homogenization temperature range is determined to be down to approximately 0°C, while the unreformed A lower limit of about 22°C would have been generated as homogeneity. Based on this more precise prediction, users can better select promising refrigerant mixtures before running tests in the laboratory.

[0075] Figure 8 880 is a simplified block diagram of a computer-based system 880 that can be used to simulate liquid mixtures according to any type of embodiments of the present invention described herein. System 880 includes a bus 883. Bus 883 is used as an interconnection between the various components of system 880. Connected to bus 883 is an input / output device interface 886, which is used to connect various input and output devices (e.g., keyboard, mouse, touch screen, display, speaker, etc.) to system 880. Central processing unit (CPU) 882 is connected to bus 883 and provides the execution of computer instructions. Memory 885 provides volatile storage for data used to execute computer instructions. Storage device 884 provides non-volatile storage for software instructions such as operating system (not shown). System 880 also includes a network interface 881, for connecting to any type of network known in the art, including wide area network (WAN) and local area network (LAN).

[0076] It should be understood that the example embodiments described herein can be implemented in many different ways. In some cases, the various methods and machines described herein can be implemented by a physical, virtual, or hybrid general-purpose computer (e.g., computer system 880) or computer network environment (e.g., computer environment 990, as described below with respect to Fig. 9 The computer system 880 may be converted into a machine that performs the methods described herein, for example, by loading software instructions that implement the method 220 into the memory 885 or non-volatile storage 884 for execution by the CPU 882. It should be further understood by those of ordinary skill in the art that the system 880 and its various components may be configured to perform any embodiment or combination of embodiments of the present invention described herein. In addition, the system 880 may implement the various embodiments described herein using any combination of hardware, software, and firmware modules that are operably coupled to the system 880 internally or externally.

[0077] Fig. 9A computer network environment 990 is illustrated in which embodiments of the present invention may be implemented. In the computer network environment 990, a server 991 is linked to clients 993a-n via a communication network 992. The environment 990 may be used to allow clients 993a-n (alone or in combination with server 991) to perform any of the embodiments described herein. For non-limiting examples, the computer network environment 990 provides cloud computing embodiments, software as a service (SAAS) embodiments, and the like.

[0078] The embodiments or aspects thereof may be implemented in the form of hardware, firmware or software. If implemented in software, the software may be stored on any non-transitory computer-readable medium configured to enable a processor to load the software or a subset of its instructions. The processor then executes the instructions and is configured to operate or cause the device to operate in the manner described herein.

[0079] In addition, firmware, software, routines, or instructions may be described herein as performing certain actions and / or functions of a data processor. However, it should be appreciated that such descriptions are included herein only for convenience, and such actions are actually produced by a computing device, processor, controller, or other device that executes the firmware, software, routines, instructions, etc.

[0080] It should be understood that flow charts, block diagrams, and network diagrams may include more or fewer elements, be arranged differently, or be represented differently. However, it should also be understood that certain embodiments may specify that the number of block diagrams and network diagrams and block diagrams and network diagrams illustrating the execution of an embodiment be implemented in a specific manner.

[0081] Thus, further embodiments may also be implemented in various computer architectures, physical computers, virtual computers, cloud computers, and / or some combination thereof, and therefore, the data processors described herein are for illustration purposes only and not as a limitation of the embodiments.

[0082] The teachings of all patents, published applications, and references cited herein are incorporated by reference in their entirety.

[0083] While example embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and details may be made without departing from the scope of the embodiments as covered by the following claims.

[0084] References

[0085] [1]Wilson,K.G.Renormalization group and critical phenomena.II.Phase-space cell analysis of critical behavior.Phys.Rev.B 1971,4,3184.

[0086] [2]Hohenberg,P.C.,and A.P.Krekhov."An introduction to the Ginzburg–Landau theory of phase transitions and nonequilibrium patterns."PhysicsReports 572(2015):1-42.

[0087] [3]Walker,James S.,and Chester A.Vause."Lattice theory of binaryfluid mixtures:Phase diagrams with upper and lower critical solution pointsfrom a renormalization-group calculation."The Journal of chemical physics79.6(1983):2660-2676.

[0088] [4]Yu,Fan,and Jun Cai."Renormalization Group Approach to BinaryLiquid–Liquid Equilibria."Industrial&Engineering Chemistry Research 59.20(2020):9611-9618.

[0089] [5]Hahn,Maximilian,and Klamt,Andreas.“Lattice-Monte simulations nearthe critical point”,to be published

[0090] [6]Klamt,Andreas,Gerard JP Krooshof,and Ross Taylor."COSMOSPACE:Alternative to conventional activity-coefficient models."AIChE journal 48.10(2002):2332-2349.

[0091] [7]Lee,D.D.,J.H.Choy,and J.K.Lee."Computer generation of binary andternary phase diagrams via a convex hull method."Journal of phase equilibria13.4(1992):365-372.

Claims

1. A computer-implemented method for simulating a liquid mixture, the method comprising: receiving a free energy profile of a liquid mixture comprising two components; constructing a partition function describing the fluctuation of the mole fractions of the two components in the liquid mixture based on the received free energy curve, wherein the constructed partition function covers symmetric and asymmetric fluctuations of the mole fractions of the two components around an average mole fraction; Using the constructed partition function, calculating a renormalized free energy curve of the liquid mixture; and Using the calculated renormalized free energy curve, the behavior of the liquid mixture was simulated.

2. The method of claim 1, wherein: The renormalized free energy curve is the constructed partition function multiplied by the negative logarithm of the molar thermal energy RT.

3. The method of claim 1, wherein: The renormalized free energy curve is a function of a temperature scale parameter.

4. The method of claim 3, wherein: The temperature scale parameter is a constant or a function of a property determined based on the received free energy curve of the liquid mixture.

5. The method of claim 1, wherein: Using the calculated renormalized free energy curve to simulate the behavior of the liquid mixture includes: Using the tangent construction method, the liquid-liquid equilibrium of the liquid mixture is predicted.

6. The method of claim 1, wherein: Using the calculated renormalized free energy curve to simulate the behavior of the liquid mixture includes predicting at least one of the following: Critical solution temperature; Lower critical solution temperature; Upper critical solution temperature; and The renormalized equilibrium composition of the two components.

7. A system for simulating a liquid mixture, the system comprising: processor; as well as a memory having computer code instructions stored thereon, with which the processor and the memory are configured to cause the system to: receiving a free energy profile of a liquid mixture comprising two components; constructing a partition function describing the fluctuation of the mole fractions of the two components in the liquid mixture based on the received free energy curve, wherein the constructed partition function covers symmetric and asymmetric fluctuations of the mole fractions of the two components around an average mole fraction; Using the constructed partition function, calculating a renormalized free energy curve of the liquid mixture; and Using the calculated renormalized free energy curve, the behavior of the liquid mixture was simulated.

8. The system of claim 7, wherein: The renormalized free energy curve is the constructed partition function multiplied by the negative logarithm of the molar thermal energy RT.

9. The system of claim 7, wherein: The renormalized free energy curve is a function of a temperature scale parameter.

10. The system of claim 9, wherein: The temperature scale parameter is a constant or a function of a property determined based on the received free energy curve of the liquid mixture.

11. The system of claim 7, wherein: To simulate the behavior of the liquid mixture using the calculated renormalized free energy curve, the processor and the memory are further configured to cause the system to: Using the tangent construction method, the liquid-liquid equilibrium of the liquid mixture is predicted.

12. The system of claim 7, wherein: To simulate the behavior of the liquid mixture using the calculated renormalized free energy curve, the processor and the memory are further configured, using the computer code instructions, to cause the system to predict at least one of the following: Critical solution temperature; Lower critical solution temperature; Upper critical solution temperature; and The renormalized equilibrium composition of the two components.

13. A non-transitory computer program product for simulating a liquid mixture, the computer program product being executed by a server in communication with one or more clients across a network, and comprising: A computer readable medium comprising program instructions which, when executed by a processor, cause the processor to: receiving a free energy profile of a liquid mixture comprising two components; constructing a partition function describing the fluctuation of the mole fractions of the two components in the liquid mixture based on the received free energy curve, wherein the constructed partition function covers symmetric and asymmetric fluctuations of the mole fractions of the two components around an average mole fraction; Using the constructed partition function, calculating a renormalized free energy curve of the liquid mixture; and Using the calculated renormalized free energy curve, the behavior of the liquid mixture was simulated.

14. The computer program product of claim 13, wherein: The renormalized free energy curve is the constructed partition function multiplied by the negative logarithm of the molar thermal energy RT.

15. The computer program product of claim 13, wherein: The renormalized free energy curve is a function of a temperature scale parameter.

16. The computer program product of claim 15, wherein: The temperature scale parameter is a constant or a function of a property determined based on the received free energy curve of the liquid mixture.

17. The computer program product of claim 13, wherein: To simulate the behavior of the liquid mixture using the calculated renormalized free energy curve, the program instructions cause the processor to: Using the tangent construction method, the liquid-liquid equilibrium of the liquid mixture is predicted.

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