A new one-site coarse-grained model for water: Bottom-up many-body projected water (BUMPer). II. Temperature transferability and structural properties at low temperature

Abstract

A number of studies have constructed coarse-grained (CG) models of water to understand its anomalous properties. Most of these properties emerge at low temperatures, and an accurate CG model needs to be applicable to these low-temperature ranges. However, direct use of CG models parameterized from other temperatures, e.g., room temperature, encounters a problem known as transferability, as the CG potential essentially follows the form of the many-body CG free energy function. Therefore, temperature-dependent changes to CG interactions must be accounted for. The collective behavior of water at low temperature is generally a many-body process, which often motivates the use of expensive many-body terms in the CG interactions. To surmount the aforementioned problems, we apply the Bottom-Up Many-Body Projected Water (BUMPer) CG model constructed from Paper I to study the low-temperature behavior of water. We report for the first time that the embedded three-body interaction enables BUMPer, despite its pairwise form, to capture the growth of ice at the ice/water interface with corroborating many-body correlations during the crystal growth. Furthermore, we propose temperature transferable BUMPer models that are indirectly constructed from the free energy decomposition scheme. Changes in CG interactions and corresponding structures are faithfully recapitulated by this framework. We further extend BUMPer to examine its ability to predict the structure, density, and diffusion anomalies by employing an alternative analysis based on structural correlations and pairwise potential forms to predict such anomalies. The presented analysis highlights the existence of these anomalies in the low-temperature regime and overcomes potential transferability problems.

I. INTRODUCTION

As a marriage of statistical mechanical theory and computation, computer simulations have been extensively employed to simulate both the chemical and physical behavior of many systems of interest.^1–9 Computer simulations have enabled these explorations from an atomistic point of view, although the extent of these atomistic simulations is limited by computational time and cost. For example, using atomistic force fields, it is computationally infeasible to fully observe rare events, such as protein folding, which require larger spatiotemporal scales.^10,11 Even for smaller systems, this limitation has been a bottleneck to explore phenomena with much larger activation energies than thermal energy (k_BT), i.e., glassy dynamics or nucleation, without utilizing advanced sampling techniques.¹²

Coarse-grained (CG) models have recently gained attention as a way to address the computational bottlenecks. Specifically, CG models can greatly enhance the accessible spatiotemporal scales of computer simulations by integrating out or somehow averaging over unnecessary or fast degrees of freedom.^11,13–20 In order to retain essential physics observed at the microscopic resolution, the so-called “bottom-up” CG methods and the corresponding CG models have been developed so that the CG model still conserves structural correlations observed in the fine-grained (FG) system (e.g., atomistic).^{18,19,21–23} By doing so, various studies have shed light on complex collective behavior where conventional atomistic simulations could not (for some examples from the study of HIV-1, see Refs. 24–28)

As bottom-up CG models have successfully been used to simulate larger spatial scale systems, we can attempt to leverage these advances in CG modeling to explore the structural changes and phase transitions in the low-temperature regime. Due to the many-body correlations emergent in structural transformations and high energy barriers, conventional atomistic simulations have been challenging in this regime, and even experimentalists often find it problematic to perform experiments that are stable enough under such low-temperature conditions.^29–33 Therefore, carefully designed bottom-up CG models are needed to elucidate the mechanisms underlying many low-temperature phenomena. An important consideration when designing bottom-up CG models is their transferability.^{18,21–23,34} Since these CG models are carefully parameterized from reference FG systems, the resultant bottom-up CG model is not guaranteed to faithfully represent thermodynamic state points that were not used in their parameterization. On a related point, it has been shown that, arguably, the most rigorous bottom-up CG effective potential is equivalent to the many-body potential of mean force (PMF) for the CG variables under the condition of thermodynamic consistency.^35,36 As the many-body PMF can be thought of as a configuration-dependent free energy function, the CG interaction from specific state points, such as the Helmholtz free energy under constant NVT, cannot be transferred to different temperatures or pressures without an explicit state variable dependence. This issue is known as the transferability problem.^{19,23,34,37,38} Although bottom-up CG models have been able to faithfully reproduce structural correlations, the transferability issue has hindered their applicability to lower temperatures.

Rather than reparameterizing CG models for each state point of interest, we recently revisited the concept of free energy decomposition^39,40 to allow for a flexible model that can explore different temperatures within the canonical ensemble. Because the Helmholtz free energy is decomposed into energetic and entropic contributions, ΔW_CG(R) = ΔU_CG(R) − TΔS_CG(R),⁴¹ we explicitly extracted the CG energetic, ΔU_CG(R), and entropic, −TΔS_CG(R), contributions that allowed a single CG model to encompass different temperatures.³⁸ Using this strategy, we seek here to examine the low-temperature behavior of liquids based on a bottom CG model designed at room temperature. Specifically, water is of particular interest to us, since the abnormal structural changes and nucleation processes of water are argued to be important for both certain biological function and industrial processes.^42–50 In this light, several top-down CG models, including the mW model,^51,52 have been designed to probe collective behavior of water in the low-temperature regime.^53–59 However, relatively less attention has been paid to systematic studies using bottom-up CG models that can rigorously recapitulate the structural behavior with respect to the FG system.

The abnormal changes in water at low temperatures are manifested by a variety of anomalies; namely, structure, diffusion, and density anomalies. We denote them as hierarchical anomalies given their pronounced hierarchy in the context of pressure–temperature phase behavior.⁶⁰ The density anomaly is perhaps the most well-known and implies the existence of a region called the temperature of maximum density, where the density decreases with the decrease in temperature under constant pressure.⁶¹ The diffusion anomaly is related to the unusual behavior of the diffusion constant D, which increases during compression at low temperatures, contradicting the usual behavior observed in normal liquids from both the experiment⁶¹ and simulation.^60,62–64 Finally, the structure anomaly refers to the existence of specific regions where liquid becomes more disordered, e.g., as characterized by order parameters, compared to higher density conditions. Along with nucleation, it would be of great interest to simulate these anomalies with systematic bottom-up CG models. This work represents a first step in that direction.

In this second paper of the Bottom-Up Many-Body Projected Water (BUMPer) series, we aim to address temperature transferability and explore low-temperature behavior of water using our many-body projected CG model. In the preceding paper⁶⁵ (hereafter referred to as “Paper I”), we were able to effectively project three-body interactions onto pairwise basis sets to construct the BUMPer model. Despite using pairwise basis sets, BUMPer was shown to reproduce two-, three-, and many-body correlations at room temperature better than other bottom-up CG models. As a natural extension, we now examine if this pairwise water model can also capture nucleation behavior at low temperature. More importantly, it is informative to analyze if BUMPer exhibits the anomalous behaviors at even lower temperatures that have been observed in both theory and experiment.

The remainder of this paper is organized as follows: In Sec. II, we briefly review the many-body projection theory and resultant BUMPer models. After discussing the free energy decomposition of the many-body CG PMF, we address temperature transferability in Sec. III and construct a temperature transferable CG model for bulk water. We further extend this transferability to simulate an ice/water interface at the CG level and explore if BUMPer enables freezing at the interface. Systematic order parameters on the basis of local homogeneity in a molecular environment are designed and applied to this system. Finally, to conclude, we demonstrate transferability of a BUMPer model that is parameterized at room temperature to much lower temperatures in order to assess its ability to manifest some of the aforementioned anomalous properties.

II. THEORY

A. Review: Many-body projection theory and the BUMPer model

The physical origin of BUMPer stems from a many-body projection theory that faithfully projects many-body interaction terms onto lower-order basis sets.⁶⁵ As extensively discussed in Paper I, the main idea of this theory shares physical intuition from liquid state physics, namely, the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy.⁶⁶ In practice, we utilize the generalized many-body projection theory for three-body interactions. The many-body expansion (MBE) of the many-body CG PMF into a corresponding n-body basis set gives⁶⁷

$$\begin{array}{ccc}\hfill U& =\hfill & \sum _I\sum _{J\ne I}{U}^{\left(2\right)}\left(R_{IJ}\right)+\sum _I\sum _{J\ne I}\sum _{K>J}{U}^{\left(3\right)}\left({\theta }_{JIK},R_{IJ},R_{IK}\right)\hfill \\ \hfill & \hfill & +\sum _{IJKL}{U}^{\left(4\right)}\left({\theta }_1,{\theta }_2,\varphi \right)+\cdots ,\hfill \end{array}$$ (1)

where U⁽ⁿ⁾ denotes the n-body interaction potential with the appropriate basis sets, such as the pairwise basis sets R_IJ for two-body interactions, the triplet angle θ_JIK and pair distances R_IJ and R_IK for three-body interactions, and so on. For the case of water, we ignore contributions higher than three-body because strong hydrogen bonding can be regarded as a result of three-body interactions at large enough temperatures.⁶⁸ Inspired by a previously reported computer modeling of water, we specifically utilize the Stillinger–Weber (SW) interaction⁶⁹ with the following form:

$$\begin{array}{ccc}\hfill U^{\left(3\right)}\left({\theta }_{JIK},R_{IJ},R_{IK}\right)& =\hfill & {\lambda }_{JIK}{\left(\text{cos\hspace{0.17em}}{\theta }_{JIK}-\text{cos\hspace{0.17em}}{\theta }_0\right)}^2\hfill \\ \hfill & \hfill & \times \text{\hspace{0.17em}exp}\left(\frac{{\gamma }_{IJ}}{R_{IJ}-a_{IJ}}\right)\text{exp}\left(\frac{{\gamma }_{IK}}{R_{IK}-a_{IK}}\right),\hfill \end{array}$$ (2)

where λ_JIK is the interaction strength (in kcal/mol) and a_IJ is the cutoff distance (in Å). From Eq. (2), one can construct the CG model with the explicit three-body interactions. Specifically, in Paper I,⁶⁵ we utilized force-matching to construct the three-body force-matched (3B-FM) CG model based on Ref. 70. Yet, having explicit three-body interactions in CG models results in larger computational costs during parameterization and simulation. Therefore, the main objective of many-body projection theory is to construct a faithful representation of three-body interactions using only pairwise basis dependent on R_IJ in this case. In other words, an analytic formulation of the projected interaction should only be dependent on R_IJ such that

$$\begin{array}{ccc}\hfill U_{eff}^{\left(2\right)}\left(R_{IJ}\right)& =\hfill & \sum _{K>J}{\lambda }_{JIK}{\left(\text{cos\hspace{0.17em}}{\theta }_{JIK}-\text{cos\hspace{0.17em}}{\theta }_0\right)}^2\hfill \\ \hfill & \hfill & \times \text{\hspace{0.17em}exp}\left(\frac{{\gamma }_{IJ}}{R_{IJ}-a_{IJ}}\right)\text{exp}\left(\frac{{\gamma }_{IK}}{R_{IK}-a_{IK}}\right).\hfill \end{array}$$ (3)

We developed a general numerical framework that generates the projected interaction from Eq. (3) by constructing a conditional average at the fixed distance R_IJ: ${⟨U^{\left(3\right)}\left({\theta }_{JIK},R_{IJ},R_{IK}\right)⟩}_{p\left({\theta }_{JIK},R_{IK}\text{|}{R}_{IJ}\right)}.$

The final CG potential, named the “projected three-body force-matched” (P3B-FM) CG model, based on Eq. (1) includes the two-body contribution $U_{3b}^{\left(2\right)}\left(R_{IJ}\right)$ and the projected interaction $U_{eff}^{\left(2\right)}\left(R_{IJ}\right)$, as given in the following equation, which is illustrated by the schematic shown in Fig. 1:

$$\begin{array}{ccc}\hfill U\left({\mathbf{R}}^N\right)& =\hfill & \sum _I\sum _{J\ne I}\left\{U_{3b}^{\left(2\right)}\left(R_{IJ}\right)+2\left(N_c-1\right)\right.\hfill \\ \hfill & \hfill & \times \left.\int d{\theta }_{JIK}d{R}_{IK}p\left({\theta }_{JIK},R_{IK}\text{|}{R}_{IJ}\right)U_{3b}^{\left(3\right)}\left({\theta }_{JIK},R_{IJ},R_{IK}\right)\right\}.\hfill \end{array}$$ (4)

Because the three-body interactions in water have a finite cutoff, the long-range structure of the P3B-FM CG model beyond this cutoff should be corrected. In practice, we correct this behavior by iteratively force-matching⁷¹ the P3B-FM CG model as the first iteration. Therefore, throughout this series, we denote the projected CG interaction with a correct long-range description as the BUMPer model. In Paper I of the series, we demonstrated that the BUMPer CG model can capture pairwise and many-body structural correlations correctly using only pairwise interactions (see Paper I for further theory and analyses).⁶⁵

FIG. 1.

Schematic diagram of the many-body projection theory used to construct the CG models studied in this work with their nomenclatures. In contrast to the conventional SP-FM CG model, the many-body projection theory consists of three consecutive steps. (1) Carry out three-body force matching to obtain both the two-body interaction and three-body interaction force fields, resulting in the 3B-FM CG model. (2) Based on the conditional probability histogram, three-body interactions are smeared into the two-body (pairwise) basis sets. Then, the P3B-FM CG model is constructed by summing the projected three-body interaction and original two-body interaction from the three-body force-matching. (3) Finally, the BUMPer CG model corresponds to this projected CG model with corrected long-range interactions beyond the three-body interaction cutoff.

B. Free energy decomposition of the CG many-body PMF

Bottom-up CG approaches aim to derive CG interactions that satisfy the thermodynamic consistency relationship between FG and CG ensembles.³⁵ The resultant CG interaction is known to follow a certain form,

$$\begin{array}{ccc}\hfill U\left({\mathbf{R}}^N\right)& =\hfill & -k_{B}T\text{\hspace{0.17em}ln}\int d{\mathbf{r}}^n\text{\hspace{0.17em}exp}\left(-\frac{u\left({\mathbf{r}}^n\right)}{k_{B}T}\right)\hfill \\ \hfill & \hfill & \times \delta \left({\text{M}}_{\mathbf{R}}^N\left({\mathbf{r}}^n\right)-{\mathbf{R}}^N\right)+\left(\text{const.}\right),\hfill \end{array}$$ (5)

where the constant term is not dependent on R^N and the delta function notation here is understood to be a product of delta functions for each CG mapping ${\text{M}}_{\mathbf{R}}^N\left({\mathbf{r}}^n\right):{\prod }_{I=1}^N\delta \left({\text{M}}_{\mathbf{R}}\left({\mathbf{r}}^n\right)-{\mathbf{R}}_I\right)$. Equation (5) indicates that the CG potential approximates the many-body CG PMF, U(R^N), which is a free energy quantity. In other words, the many-body CG PMF is the configurational PMF conditioned based on the CG mapping functions ${\text{M}}_{\mathbf{R}}^N\left({\mathbf{r}}^n\right)$. As such, we can draw upon fundamental relationships from statistical mechanics that further decompose the free energy function into energetic and entropic contributions. Because the CG PMF is often spanned by pairwise basis sets, the pairwise approximation provides the following decomposition in the canonical ensemble:

$$\begin{array}{c}\hfill U\left(R;T\right)=\mathrm{\Delta }{E}_{\text{CG}}\left(R\right)-T\mathrm{\Delta }{S}_{\text{CG}}\left(R\right).\hfill \end{array}$$ (6)

In this section, we introduce the symbol Δ to refer to the thermodynamic quantity at R in reference to R → ∞. This Helmholtz free energy decomposition scheme provides a simple yet physical way to help understand the transferability issue with respect to temperature and even other features of the system.

C. Representability problems

We note that numerous efforts have been undertaken to compute various thermodynamic properties of water, e.g., heat capacity, heat of vaporization, dielectric constant, molecular quadruple, and isothermal compressibility, using different water models and to quantitatively evaluate the performance of these models compared to experimental values.^72–87 However, for CG systems, the representability issue indicates that calculating certain thermodynamic properties of CG models requires special care due to differences in observable expressions between the FG and CG ensembles.^19,23,37,38 Thus, carefully designed methods that aim to provide CG observables compatible with the underlying FG systems must be pursued.⁸⁸ As a first step, however, we focus on thermodynamic expressions of the bottom-up CG model that can be expressed in terms of the CG PMF representation. Subsequently, follow-up work will be carried out to obtain compatible CG observable expressions to fill this gap, e.g., configurational entropy,^38,89 pressure,⁸⁸ and internal energy,^90,91

D. Simulation details

All-atom (FG) simulations performed in this work were based on the protocol developed in Paper I.⁶⁵ In brief, we conducted atomistic simulations using four different water models (other force fields can be applied without modification): Simple Point Charge/Extended (SPC/E),⁹² Simple Point Charge/Flexible (SPC/Fw),⁹³ Transferable Intermolecular Potential with 4 Points version 2005 (TIP4P/2005),⁹⁴ and TIP4P/ice⁹⁵ force fields. In Paper I,⁶⁵ we only constructed a BUMPer model at T = 300 K. In order to construct a temperature transferable model under Helmholtz free energy conditions (canonical ensemble constant NVT conditions),³⁹ we generated the CG systems with different temperatures by fixing the box dimension from the equilibration step at 300 K and adjusting the final temperature in constant NVT dynamics with an initial velocity based on the target temperature. For each water model, we conducted five simulations at different temperatures with an equal temperature spacing of ΔT = 20 K: 280 K, 300 K, 320 K, 340 K, and 360 K.

Using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) molecular dynamics (MD) engine, we generated the atomistic trajectories as follows: Starting from the last snapshot of the SPC/Fw water model at T = 300 K with an equilibrium density of 0.978 g/cm³, we ran constant NVT dynamics at each desired temperature with a Nosé–Hoover thermostat^96,97 with a coupling constant of τ_NVT = 0.1 ps. To achieve sufficient sampling to construct the corresponding CG model, we collected trajectory data every 0.25 ps during the NVT run for an overall time of 5 ns. From the CG mapped FG trajectories, several CG models were parameterized via different force-matching schemes; procedural details are discussed in Paper I.⁶⁵

III. RESULTS

A. Temperature transferability

1. BUMPer

Figure 2 summarizes the temperature transferability analysis in which fitted CG models on the basis of a linear dependence on temperature are shown to match each CG model as the temperature is varied, thereby also validating the applicability of the proposed decomposition scheme.^38–40 In other words, we designed the actual BUMPer model to be transferable to different temperatures in the liquid phase. Based on the previous work and Eq. (6) introduced above,^38–40 this is done by first obtaining the pairwise CG interaction functions [ΔE_CG(R) and ΔS_CG(R)] from different reference points, after which interpolation to different temperatures is possible. By assuming that the pairwise thermodynamic functions are invariant over different temperatures within the same phase,³⁸ we employed a two-point interpolation scheme using finite differences at T = 300 K (reference) and 360 K (the highest simulated temperature for the liquid phase) in order to maximize the efficiency of parameterization. For the sake of simplicity, we report the results from the TIP4P/2005 force field here. Detailed instructions for the construction of BUMPer at desired temperatures from different force fields are available in a Github repository.⁹⁸

FIG. 2.

Temperature transferability of BUMPer assessed by the pair structures and CG PMFs at different temperatures ranging from 280 K (red) to 360 K (blue). (a) Effect of temperature on the g(R) functions obtained from the fitted CG model, implying that a non-temperature-dependent CG potential function is not able to capture differences in the pair structure. (b) Temperature dependence of the local density distribution (N-body correlation) of atomistic (solid lines) and fitted CG (dashed lines) models ranging from 280 K (red) to 360 K (blue). (c) Effective CG PMFs of the BUMPer models for different temperatures obtained using the computational protocol described in Fig. 1 (lines) and CG PMFs interpolated using Eq. (6) (dots).

The high fidelity features of the derived BUMPer models are evaluated by computing their radial distribution and local density distribution functions, as shown in Figs. 2(a) and 2(b), respectively. To compute the local density distribution from the simulation trajectories, we followed the definition used in Paper I:⁶⁵ ${\rho }_I={\sum }_{I,J\text{\hspace{0.17em}neigh}}\frac{1}{2}\left(1+\text{tanh}\left(\frac{R_{IJ}/R_c-1}{{\sigma }_R}\right)\right)$, with R_c = 4.5 Å and σ_R = 0.1. Although we found that local density correlations are less sensitive to temperature than pair correlations, the extrapolated model correctly captures the width and intensity of each distribution in the reported temperature range. As expected, water pair correlations become softer at higher temperatures, but BUMPer recapitulates the pair correlations at these varied temperatures. Analytically fitted CG PMFs at different temperatures ranging from 280 K to 360 K at 20 K intervals are also shown in Fig. 2(c) and compared to the parameterized CG PMFs (references) at each target temperature. Notably, the fitted BUMPer PMFs are almost identical to their reference profiles at each temperature. It is also worth noting that we only utilized two temperatures, 300 K and 360 K, to not only interpolate but also extrapolate the CG PMFs to temperatures as low as 280 K, which were not included in our training set.

2. Temperature transferability of other CG models

It is evident from Eqs. (5) and (6) that the CG PMFs on the basis of pairwise distances vary over different temperatures and, thus, it is not physically correct to use a non-temperature-dependent CG force field in this basis to span different temperatures; by doing so, for example, one may neglect the entropic effects that are embedded in the CG PMF. To demonstrate this temperature transferability issue more quantitatively, we compared our temperature transferable BUMPer results with a single temperature BUMPer CG model that is parameterized at T = 300 K and then naïvely simulated at other temperatures ranging from 280 K to 360 K without accounting for temperature related changes. Without loss of generality, we chose the TIP4P/2005 force field in this section, but the procedure can be applied to any force field studied in this work.

For each temperature except 300 K, we calculated the water–water radial distribution function (RDF), g(R), for the naïve models and compared them to the CG mapped RDFs from all-atom simulations in Fig. 3. (The latter simulations are full all-atom simulations in which the CG mapping operators are tracked and averaged as a function of the all-atom variables, i.e., they are the “exact” result for a given FG water model.) Noticeable deviations are observed at the first peak, as well as the first minimum and even in the long-range regime. We attribute the failure of the naïve temperature model to the fact that the CG potential is no longer an effective approximation of the free energy functional at these different state points. However, one can repair this deviation by constructing BUMPer with explicit temperature dependence in the entropic term using the energy–entropy decomposition described in Sec. II B.

FIG. 3.

The assessment of the transferability of the single (naïve) BUMPer force field at the fixed temperature (T = 300 K). Using BUMPer parameterized at T = 300 K, we performed the four different simulations at 280 K, 320 K, 340 K, and 360 K. (a)–(d) illustrate the intermolecular CG RDF for each temperature, respectively. It is immediately evident that the single temperature BUMPer model gives wrong pair structures at other temperatures compared to the CG mapped FG simulations, indicating poor transferability.

Beyond the aforementioned bottom-up CG models, we also investigated the (1) temperature transferability and (2) general interaction profile of the mW model projected onto pairwise basis sets. We note that the mW model is not designed to match the many-body CG PMF, and temperature transferability is not guaranteed. Still, it is of interest to assess the similarity of the BUMPer and projected pairwise interactions from mW since both models are based on a SW-type three-body interaction. To compare mW and BUMPer on the basis of pairwise interactions, we first carried out an mW CG simulation, which uses the following interaction form:⁵²

$$\begin{array}{ccc}\hfill U_{\text{mW}}\left(R_{IJ}\right)& =\hfill & \sum _{J>I}\left\{Aϵ\left[B{\left(\frac{\sigma }{R_{IJ}}\right)}^4-1\right]\text{exp}\left(\frac{\sigma }{R_{IJ}-a\sigma }\right)\right\}\hfill \\ \hfill & \hfill & +\sum _{J\ne I}\sum _{K>J}\left\{\lambda ϵ{\left[\text{cos\hspace{0.17em}}{\theta }_{IJK}-\text{cos\hspace{0.17em}}{\theta }_0\right]}^2\right.\hfill \\ \hfill & \hfill & \times \left.\text{exp}\left(\frac{\gamma \sigma }{R_{IJ}-a\sigma }\right)\text{exp}\left(\frac{\gamma \sigma }{R_{IK}-a\sigma }\right)\right\},\hfill \end{array}$$ (7)

with parameters A = 7.049 556 277, B = 0.602 224 558 4, θ₀ = 109.47°, ϵ = 6.189 kcal/mol, σ = 2.3925 Å, a = 1.8, λ = 23.15, and γ = 1.2 from the original paper.⁵² Using the mW trajectories, we applied the MS-CG framework to obtain “re-coarse-grained” (re-CG) effective two-body CG interaction parameters. By doing so, we generated a re-parameterized top-down CG model using bottom-up algorithms. Although the re-parameterized CG model may not impart the same physical phenomena that we would expect, we assumed that the re-CG process can still recapitulate the important structural correlations emergent in the top-down CG model. Following the previous discussion, we generated two re-CG models: the simple pairwise mW (SP-mW) model and the “mW-BUMPer” model from iterative force-matching. All system sizes and temperature conditions are the same as the conventional BUMPer utilized in this work. The resultant CG potentials from the two re-CG models are illustrated in Fig. 4. Compared to other pairwise CG models of water, the potential profiles shown in Fig. 4(a) exhibit notable differences. First, changes in the CG PMF over different temperatures are negligibly small. Second, the characteristic local minimum near 2.5 Å–3 Å, where water exhibits a strong peak in pairwise correlations, is missing, which will likely affect the fidelity of the liquid structure. This anomaly is removed in mW-BUMPer, as shown in Fig. 4(b), where the potential profile captures two local minima below 5 Å, although the depths of these minima remain different from that of other BUMPer models.

FIG. 4.

Temperature transferability of the CG models based on the mW force field by comparing the CG PMFs at different temperatures ranging from 280 K (red) to 360 K (blue). (a) Effective CG PMFs of the SP-mW CG model. (b) Effective CG PMFs of the mW-BUMPer CG model.

We next evaluated explicit differences between the mW-BUMPer potential in Fig. 4(b) and the BUMPer potential in Fig. 2(c). The most significant difference is that the mW-BUMPer potential has a small bump after the first minimum, even though the value (from −0.6 kcal/mol to −0.5 kcal/mol) and location (from 2.9 Å to 2.95 Å) for the first minimum are relatively well-captured. This bump is indicative of a microscopic inconsistency in terms of structural correlations, as pairwise CG water models are often approximated as isotropic CG potentials with two characteristic scales, which emerge from hydrogen bonding and van der Waals interaction distances;⁹⁹ further discussion on the interaction profile of water will be addressed in Sec. III C. Particularly, the long-range interactions after the first peak minimally change with temperature in contrast to BUMPer shown in Fig. 2(c). This suggests that mW-BUMPer may miss certain long-range correlations exhibited by water. Nevertheless, mW-BUMPer can modulate changes in the PMF under different temperatures, especially at the first minimum. The general improvement as shown in Fig. 4(a) to Fig. 4(b) highlights the applicability of the present theory.

A last thing to note is the magnitude of the temperature derivative of the CG PMF. We know that the entropy that might seem lost during the CG procedure should be incorporated into the pairwise CG PMF in a rigorous bottom-up approach. Thus, the ensemble average of the temperature derivative should characterize the corresponding CG mapping entropy.^38,40,89 Note that the entropic component of the CG PMF is configuration-dependent −TΔS(R), but its average over the CG configurations is shown to correspond to the missing entropic quantity during the CG process (see Refs. 38, 40, and 89 for detailed theoretical derivations and discussions). From Fig. 4(b), the changes to the mW-BUMPer PMFs at the first minimum are about $\frac{0.20\text{\hspace{0.17em}kcal}/\text{mol}}{80\text{\hspace{0.17em}K}}\approx 2.5\text{\hspace{0.17em}cal}/\text{mol}/\text{K}$, which are smaller than $\frac{\mathrm{\Delta }U}{\mathrm{\Delta }T}=\frac{0.30\text{\hspace{0.17em}kcal}/\text{mol}}{80\text{\hspace{0.17em}K}}\approx 3.75\text{\hspace{0.17em}cal}/\text{mol}/\text{K}$ from the BUMPer PMFs shown in Fig. 2(c). Generally speaking, the mW model seems to capture the changes in entropy that are embedded within the CG PMF, although the changes are smaller compared to BUMPer, indicating only a partial recapitulation of the missing entropy. Further detailed analysis of the missing entropy of CG water models will be pursued in a subsequent paper of this series.

B. Liquid to solid phase transition: Ice/water interface

1. Initial preparation of interface system

As seen from the apparent success of the mW model,^53–59 three-body interactions may be an essential component for modeling complex long-timescale behavior such as nucleation. However, to our knowledge, there is currently no bottom-up CG model designed to examine this behavior by appropriately addressing many-body correlations. Considering that the three-body interactions embedded in BUMPer can greatly enhance three-body correlations,⁶⁵ we further extend the range of this model here to encompass ice growth. As demonstrated by prior work on the liquid/ice interface,^100–105 an interfacial (coexistence) system allows us to accurately determine the freezing point and can also provide detailed mechanistic insight into the underlying dynamics of crystal growth.^106,107 Therefore, interfacial systems provide a valuable target to test the extensibility of BUMPer to different phases. It is to be noted that we do not aim here to directly elucidate the crystallization behavior of bulk ice, which might necessitate enhanced sampling techniques to overcome the nucleation free energy barrier.¹⁰⁸

Even though we demonstrated the temperature transferability of BUMPer in Sec. III A, it should also be noted that the density (volume) transferability is much more difficult to achieve due to the pressure representability issue.^{19,23,34,37,38} Since the CG degrees of freedom are much less than that of FG, the CG pressure that is calculated using the FG virial expression will provide a non-physical pressure.^23,88 This clearly limits the application of the MS-CG-based CG PMF to different volume conditions, and one cannot accurately estimate the freezing point or other relevant behaviors by simply applying BUMPer to equilibrate the volumes at different temperatures. Therefore, this section should be interpreted as an examination of the ability of BUMPer to demonstrate freezing in the prototypical system (interface). A natural extension of this is to construct a fully transferable BUMPer model that can span all possible thermodynamic state points (density and temperature), along with continuous endeavors to impart density transferability,^109–113 and can determine phase transition behaviors such as the freezing point using a single CG model. Still, as a matter of fact, bottom-up CG models are constructed to recapitulate structural correlations of the corresponding FG system at a given state point; therefore, once carefully designed, it is expected that these CG models will exhibit the same freezing point and changes in density along temperature. In this light, the TIP4P/ice force field⁹⁵ was used to correctly represent the ice phase and the freezing behavior during nucleation.^114,115

On the basis of previous protocols for CG interface simulations,^116,117 we first generated bulk liquid water composed of 1024 CG water molecules using its experimental density.⁹⁵ Using the same L_x and L_y value from the bulk system, we prepared an ice system with 512 water molecules in ice Ih phase along the two simulation domains (with L_x = 31.50 Å and L_y = 29.84 Å) that cap liquid water (see Fig. 5), yielding a total of 2048 water molecules in the final system with L_z = 68.58 Å. Unlike bulk liquid water, the initial configuration of ice was obtained by employing an algorithm reported by Buch et al.¹¹⁸ that generates a nearly zero dipole moment. Each of the three components was prepared separately under constant NVT conditions and then combined to construct the initial ice/water interface configuration. We applied energy minimization using steepest descent followed by conjugate gradient minimization to a root mean square force of 10⁻⁴ kcal/mol/Å to remove any artificial stress introduced by integrating the two phases. Alternatively, one can also prepare the initial system using a spherical seed to avoid any boundary effects.¹¹⁹ We then equilibrated the system for 0.2 ns using constant NVT MD simulations at T = 249 K and, consequently, constant NPT dynamics at P = 1 atm for 0.4 ns. Finally, we collected all-atom configurations for analysis every 1 ps by running constant NVT simulations for 5 ns at the same temperature. The thermostat and barostat used in both constant NVT and NPT simulations were identical to the settings employed for the bulk systems. In this section, we note that we chose the target temperature as 249 K, which is lower than the known melting point of TIP4P/ice water, to enforce crystallization. The initial snapshot of the CG ice/water interface is depicted in Fig. 5(a).

FIG. 5.

Ice/water interface system using the BUMPer CG model. (a) Illustration of the constructed ice/water interface. (b) Intermolecular RDF of different FG/CG systems: 3B-FM (red line), BUMPer (blue line), and mW (green line) CG models for the ice/water interface and CG mapped all-atom (black dashed line) for the bulk ice Ih phase. Magnified RDFs of the first coordination shell are shown in the inset. (c) Initial snapshot of the constructed bulk ice Ih. (d) Final snapshot of the configuration of the CG ice/water using BUMPer.

We considered two different CG models in this system: a 3B-FM CG model (with explicit two-body and three-body interactions) and a BUMPer CG model. As discussed in Paper I of this series,⁶⁵ we followed the same parameterization protocol and obtained the following parameters for the interfacial system: σ_IJ = 1.0 Å, ϵ_JIK = 1.0 kcal/mol, γ_IJ = 1.2, r_cut = 3.7 Å, cos θ₀ = −0.2924, and λ_JIK = 28.0078. Note that the λ_JIK value for the interface is larger than the one from the bulk liquid at higher temperatures, indicating that the interfacial system exhibits stronger three-body interactions than bulk liquid. Correspondingly, BUMPer was also constructed by following the same parameterization protocol, as described in Secs. II A and II D. The only difference noted here is that we repeated the same protocol for the bulk TIP4P/ice water system at a low temperature (249 K) with a slightly different density condition to account for the temperature difference. Finally, all CG simulations were performed under constant NVT dynamics at T = 249 K for 10 ns with the same thermostat settings.

2. Structural correlations in BUMPer

Figure 5(b) depicts the pair correlation functions for the 3B-FM and BUMPer CG models. For comparison, we also constructed bulk ice Ih composed of 512 water molecules, where the initial structure was extracted from an orthorhombic unit cell¹¹⁸ with final dimensions of L_x = 31.50 Å, L_y = 29.84 Å, and L_z = 18.127 Å. After minimizing the energy of the initial structure to eliminate stress, the reference atomistic trajectory for ice was generated for 5 ns at the same temperature. From Fig. 5(b), it is apparent that the 3B-FM CG model can recapitulate the strong ordering that indicates the ice structure even though we note some differences, e.g., the shoulder at the second peak near 5 Å. However, the highly structured peaks near 2.7 Å, 4.7 Å, and 6.7 Å are consistent with both CG mapped atomistic distributions and neutron diffraction experiments¹²⁰ of bulk ice Ih. BUMPer seems to reproduce this ordering up to the second coordination shell within reasonable agreement, which can be understood from its fair representation of three-body correlations shown in Paper I of this series.⁶⁵ While the accuracy of BUMPer is not as pronounced as the 3B-FM CG model, this is acceptable because BUMPer is only a pairwise model. At and after the third and fourth peaks at their local maxima, BUMPer tends to be overstructured compared to 3B-FM, which tends to be similar to the reference RDF. By comparing to the mW simulations for this system, Fig. 5(b) indicates that mW follows almost identical RDF to 3B-FM after the first peak, while the first peak at 2.7 Å [see the inset of Fig. 5(b)] is less structured by 31.0%. This deviation can be understood from its top-down nature in that the interaction parameters for mW are not directly obtained from the FG system.

3. Structural correlations in simple pairwise water

As a control study, we also constructed a simple pairwise force-matched (SP-FM, see Fig. 1) CG model from the atomistic trajectories of TIP4P/ice at T = 249 K and performed a CG simulation. Here, the SP-FM CG model is readily obtained by performing MS-CG force-matching to the atomistic trajectories using pairwise basis sets, as depicted in Appendix A.¹²¹ Figure 6 shows the resulting snapshots and calculated RDFs from the SP-FM CG simulation. After 50 ns (the same time as the CG simulations above), it is unsurprising to observe that the SP-FM CG water does not have a transition to the ice phase and, rather, exhibits homogeneous fluid behavior. The corresponding RDF in Fig. 6(c) lacks the distinct peaks that are exhibited by BUMPer and the reference atomistic system.

FIG. 6.

Ice/water interface system using the SP-FM CG model. Final snapshots of the configuration of the CG ice/water interface using the SP-FM CG force field are shown: (a) front view and (b) side view. (c) Intermolecular RDF of different FG/CG systems: SP-FM CG (red line) and BUMPer CG (blue line) models for the ice/water interface and CG mapped all-atom (black dashed line) for the bulk ice Ih phase.

This comparison substantiates our initial assumption that the embedded three-body interactions are essential for modeling the crystal growth.

4. Quantitative RDF analysis

Figures 5(b) and 6(c) epitomize the success and failure of each CG model in terms of reproducing the ice-like pair correlations and, now, we aim to quantify these differences. We define the averaged unitless Euclidean distances between RDFs provided by each CG potential as

$$\begin{array}{ccc}\hfill d_{1-2}& =\hfill & {∥g_1\left(\mathbf{R}\right)-g_2\left(\mathbf{R}\right)∥}_{1-2}\hfill \\ \hfill & =\hfill & \frac{1}{{\overline{R}}_2-{\overline{R}}_1}\sqrt{\sum _{I,J}\mathrm{\Pi }\left(R_{IJ}-\frac{{\overline{R}}_1+{\overline{R}}_2}{2}\right)\cdot {\left(g_1\left(R_{IJ}\right)-g_2\left(R_{IJ}\right)\right)}^2},\hfill \end{array}$$ (8a)

$$\begin{array}{ccc}\hfill d_{>2}& =\hfill & {∥g_1\left(\mathbf{R}\right)-g_2\left(\mathbf{R}\right)∥}_{>2}\hfill \\ \hfill & =\hfill & \frac{1}{R_{\text{cut}}-{\overline{R}}_2}\sqrt{\sum _{I,J}\mathrm{\Theta }\left(R_{IJ}-{\overline{R}}_2\right)\cdot {\left(g_1\left(R_{IJ}\right)-g_2\left(R_{IJ}\right)\right)}^2}.\hfill \end{array}$$ (8b)

In Eq. (8), we introduce two different measures ∥·∥₁₋₂ and ∥·∥_>2. The first measure ∥·∥₁₋₂ denotes the averaged Euclidean distance between the CG potentials within the first ${\overline{R}}_1$ and second ${\overline{R}}_2$ coordination shells. Analytically, this can be done by introducing the normalized boxcar function ${\mathrm{\Pi }}_{{\overline{R}}_1,{\overline{R}}_2}\left(R_{IJ}\right)$.¹²² Here, ${\mathrm{\Pi }}_{a,b}\left(x\right)=\mathrm{\Theta }\left(x-a\right)-\mathrm{\Theta }\left(x-b\right)$, where $\mathrm{\Theta }$ is the Heaviside step function. Similarly, the measure ∥·∥_>2 accounts for the Euclidean distance after the second coordination shell by utilizing the Heaviside step function $\mathrm{\Theta }\left(R_{IJ}-{\overline{R}}_2\right)$. We chose ${\overline{R}}_1$ as 2.75 Å and ${\overline{R}}_2$ as 4.45 Å. In turn, both d₁₋₂ and d_>2 provide a comparable measure of how far the target CG RDF is apart from the reference RDF, as summarized in Table I; the lower the values of d₁₋₂ and d_>2, the closer the CG RDF is to that of the reference.

TABLE I.

Effective distances between the CG RDFs from this study compared with the reference RDF from the ice [Figs. 5(b) and 6(c)] using two different measures ∥·∥₁₋₂, corresponding to the regime between the first and second coordination shells, and ∥·∥_>2, corresponding to the regime after the second coordination shell.

(a) d1−2
CG model	Distance
3B-FM	0.926
BUMPer	0.832
SP-FM	0.963
mW	0.921

	(b) d>2
CG model	Distance
3B-FM	0.122
BUMPer	0.203
SP-FM	0.294
mW	0.146

Even though SP-FM gives a poor performance in describing the pair correlations up to the second coordination shell, this can be improved using BUMPer. As seen from the d₁₋₂ values, BUMPer shows as nearly good performance as the higher order 3B-FM and mW CG models. Nevertheless, Table I(b) suggests that the averaged RDF distance in the long-range regime d_>2 for BUMPer is consistently larger than that of both the 3B-FM and mW potentials. This is consistent with our observation from Fig. 5. We attribute this discrepancy in d_>2 to an inherent limitation in pairwise interactions that results in missing long-range electrostatic interactions between water dipoles (a directional interaction) that can facilitate crystallization. A lack of triplet directionality is also noticed in the final configurations of BUMPer, where a few water moieties are located as interstitial defects. Nevertheless, the overall hexagonal structure (ice Ih) is still captured using BUMPer. Additional future work will be needed to understand these subtle differences in the long-range region and to correct the interstitial defects in order to impart a finely detailed CG model.

5. Coarse-grained local structure index

The Local Structure Index (LSI) quantifies the degree of local ordering based on the radial-neighbor distribution, which aims to measure the local molecular inhomogeneity (especially with respect to the tetrahedral order). As originally suggested by Shiratani and Sasai,^123,124 the LSI is defined as the mean-squared deviation of the oxygen–oxygen radial distance between a pivot water molecule and its jth water neighbor,

$$I=\frac{1}{N}\sum _{j=1}^N{\left[{\mathrm{\Delta }}_{j+1,j}-⟨\mathrm{\Delta }⟩\right]}^2,$$ (9)

where the radial distances r_j and the index N are defined in the following order: r₁ < r₂ < ⋯ < r_j < r_j+1 < ⋯ < r_N < r_cut < r_N+1, with Δ_j+1,j ≡ r_j+1 − r_j. We adopt a previously reported cutoff distance r_cut of 3.7 Å that differentiates the first and second coordination shells;^123,124 the LSI provides a measure of ordering between these regimes. It has been suggested that the LSI can quantitatively distinguish different molecular environments in water^125–127 to account for the inherent structure underlying supercooled conditions.^128,129 An inherent structure refers to a thermal excitation-free system obtained by minimizing, or quenching, the instantaneous configurations from MD to its local minimum given by the CG potential.^130,131 Regardless of the choice of force fields, the inherent structure described by the LSI appears to delineate two distinct structures underlying supercooled water.^{128,132–135} Recently, it has been shown that the LSI can provide a bimodal distribution of high-density-liquid (HDL)-like and low-density-liquid (LDL)-like states when applied to the inherent configurations sampled by atomistic force fields.^128,129,136 This might reflect the two distinct structures of water, HDL-like and LDL-like, under supercooled conditions,¹³⁷ but we interpret this bimodality as differences in the local coordination shell ordering regardless of the more controversial phase transition scenarios.

Independent of the utilized atomistic force field, LSI statistics allow for a comprehensive examination of water under two different configurations. First, the LSI distribution of water simulated under ambient temperature and pressure is unimodal with a peak ranging from low values (HDL-like) to high values (LDL-like).^{123,124,138,139} The LSI can be used as an order parameter to indicate many-body characteristics during freezing. However, prior uses of the LSI order parameter have been based on the oxygen–oxygen distances, which is only appropriate at the FG resolution.

For the current work, we modified the definition of the original LSI order parameter. Since information about radial distances between oxygen atoms is lost in the one-site CG system, the radial distances between the center of mass (COM) of CG molecules were used as an alternative metric. We expect the difference between the all-atom LSI and CG LSI to be negligible; as m_O ≫ m_H, the oxygen–oxygen and COM–COM configurations should be almost identical. As shown in Fig. 7, oxygen–oxygen pair correlations calculated at T = 300 K and ρ = 0.978 g cm⁻³ are almost equivalent to the COM–COM pair correlations, confirming our supposition, although some differences originating from the different resolutions should be expected. Using this modified definition, we generated the inherent CG structure and corresponding LSI analysis based on the protocol described in Appendix B.

FIG. 7.

Comparison between the oxygen–oxygen pair (FG resolution) and COM–COM pair (CG resolution) distributions by utilizing the FG/CG RDF. (a)–(d) depict the oxygen–oxygen FG RDF (solid line) and COM–COM CG RDF (dashed line) for four different water force fields: (a) SPC/E, (b) SPC/Fw, (c) TIP4P/2005, and (d) TIP4P/ice. For all the models, FG/CG distributions are almost identical—confirming that the defined CG LSI parameter can be utilized in the same way as the well-known (FG) LSI parameter.

6. Coarse-grained local structure index analysis

Figure 8 illustrates the LSI distribution for water under two different conditions. Figure 8(a) shows a unimodal probability distribution P(I) for ambient liquid water using the SPC/Fw force field. As expected, BUMPer is able to reproduce the finely detailed local structure of water. The first maximum peak at 0.015 Ų also agrees with the P(I) from Ref. 140. The CG model that is parameterized using only pairwise basis sets still retains the unimodal distribution but deviates from the atomistic simulation with an overstructured peak at the first maximum. The overly structured distribution further suggests that the SP-FM CG model tends to yield a structure consistent with temperatures larger than its parameterized temperature.¹⁴⁰ We also note that the mW model can reproduce this trend within reasonable agreement despite the top-down parameterization of the model.

FIG. 8.

Probability density distributions P(I) of the LSI in different water systems at atomistic and CG resolutions: mapped atomistic (red line), SP-FM (black line), BUMPer (green line), and mW (purple line) CG models under (a) the ambient condition (regular MD configurations) in liquid water at 300 K and (b) the inherent condition (the minimum energy structures) in the ice/water interface at 249 K.

For the ice/water interface, we generated the inherent CG structures as described above to distinguish the local ordering pertinent to ice using the TIP4P/ice force field for the FG simulation and the corresponding BUMPer model for the CG simulation. In Fig. 8(b), P(I) of the LSI shows a distinct multi-modal behavior in contrast to Fig. 8(a). The trends depicted in Fig. 8(b) slightly differ from the reported values of P(I) for inherent configurations at low temperatures near 200 K–250 K.^128,136 However, we note that the previous reports mostly focused on the supercooled liquid systems, whereas the current work focuses on the ice/water interface. The large peak observed at the I = 0 Ų limit can be understood as the configurations contributed from the frozen (or nucleated) water in the system. While agreement between P^FG and P^BUMPer is limited, some important information can be still extracted by comparing these inherent distributions. First, BUMPer has a non-zero P(I = 0) value, indicating that BUMPer is capable of simulating freezing behavior. On the other hand, it is immediately evident that this is not the case for the SP-FM CG model, which does not exhibit a bimodal distribution from inherent configurations. The difference between the P^FG(I = 0) and P^BUMPer(I = 0) can be mostly attributed to the existence of defect sites or misaligned particles while cooling, as illustrated in Fig. 5(d). Although BUMPer qualitatively captures the peak near I = 0.05 Ų and the decaying pattern afterward, BUMPer has an additional small peak near I = 0.1 Ų and different peak intensities. As discussed in the RDF analysis, this is presumably due to the absence of directional interactions. Interestingly, mW shows the opposite trend: The P(I → 0) behavior can be well-captured, whereas the other correlations at I > 0.02 Ų are missing. From a similar perspective, we attribute this unimodal distribution to the strong directional interactions imposed during the simulations. These two opposite behaviors of BUMPer and mW seem to indicate that the most accurate CG modeling of the ice/water interface requires a fine tuning between the pairwise and three-body interactions.

C. Low-temperature behavior and anomalies

1. Limitation of BUMPer

After confirming the temperature transferability of BUMPer and its ability to freeze at an ice/water interface at 249 K, we now extend our analysis to lower temperatures. As stated above, we focus on an examination of hierarchical anomalies shown in water observed by both theoretical and experimental reports. In line with the analysis of BUMPer during nucleation, we hypothesize that BUMPer can contribute to the systematic understanding of these anomalies. However, a straightforward application of BUMPer is not feasible as its temperature transferability under supercooled conditions has yet to be substantiated. Pressure is another principal state variable to be considered because of its importance in the supercooled regime. As discussed in Sec. III B 1, the pressure representability problem hinders us from directly simulating the CG model under constant NPT conditions.^{19,23,34,37,38} In this work, the effective CG interaction is based on the original many-body interaction defined in Ref. 67, which has only a configuration-dependence. However, it should be noted that introducing a volume-dependent term in the overall CG potential is one way to circumvent this problem.^109,112,141 These factors indicate that we need an indirect way of assessing BUMPer rather than explicitly running simulations under supercooled conditions.

2. Low-temperature behavior extracted from the CG model: Preliminaries

Understanding phase transitions and associated anomalies of physical systems based on their governing interactions has a long history. Stell and Hemmer pioneered an analysis that relates phase transition behavior to features of the “inner” core for pairwise potential models, such as the sign or slope.^142,143 A main achievement of this earlier work was separating the pair potential into short-range and long-range components (so-called “core-softened” potential) and constructing equations of states for each component to examine critical behavior. The core-softened potential approach has been successfully applied to understand the anomalous behavior of water in terms of its potential profile.^144–146 As discussed in Paper I, BUMPer also entails two characteristic length scales witnessed from the pair interaction profile depicted in Fig. 2(c).⁶⁵ With this in mind, we now adopt design principles of the core-softened potential approach to understand the systematically designed CG model, BUMPer.

While various potential forms with different short-range and long-range interaction strengths that exhibit water-like anomalous properties have been reported, these interactions often fall into the following categories: repulsive ramp with attractive well models and attractive double-well models. Both categories stem from statistical mechanical theory in which the former is based on the Stell–Hemmer model^142,143 and the latter is from the modified Takahashi model.^147,148 Simultaneously, theoretical attempts have also been made to uncover conditions in which the core-softened model exhibits phase transitions with accompanied anomalous behavior.^149–152 Hence, by recognizing the challenges in transferable CG models, we utilize this approach to indirectly examine BUMPer’s ability to predict hierarchical anomalies.

3. Low-temperature behavior extracted from structural properties

Stanley and co-workers have found that the capability of a model to demonstrate a water-like anomaly is embedded in its structural properties. To elaborate, for core-softened models with two characteristic lengths a and b, Yan et al.¹⁵³ developed the following criteria for anomaly occurance:

$$\frac{1}{2}<\frac{a}{b}<\frac{6}{7}.$$ (10)

The ratio of characteristic scales given by Yan’s criterion was then generalized to water’s nearest neighbor shells emergent in supercooled water. In this line of work, it has been proposed that the ratio between a and b can be understood as the ratio between r₁(ρ) and r₂(ρ) at density ρ, where r_n(ρ) denotes the pair distances corresponding to the nth maximum value shown in the RDF at temperatures lower than 207 K.¹⁵⁴ In this case, the generalized criterion is reduced to

$$\frac{1}{2}<\frac{r_1\left(\rho \right)}{r_2\left(\rho \right)}<\frac{6}{7}.$$ (11)

At first glance, this approach imparts a direct and systematic correspondence between the structural anomalies of water in terms of structural characteristics, i.e., the RDF. The proposed relationship has also been shown to be related to the density derivative of the cumulative order integral of the excess entropy.¹⁵⁴

We calculated r₁(ρ) and r₂(ρ) from the propagated BUMPer trajectory to check if Eq. (11) still holds for two different systems: bulk and ice/water interface systems at 200 K. We note that both the criteria [Eqs. (10) and (11)] were designed from the continuous shouldered well (CSW) form of the potential. Nevertheless, we argue that these structural criteria are still valid since the BUMPer interaction demonstrates two characteristic length scales. The CG simulation of supercooled water at 200 K was carried out using the interaction depicted in Fig. 2(c). For the ice/water interface simulation, to assure the temperature condition from Eq. (11), we re-parameterized the CG interaction of the interface system at 200 K from the original temperature of 249 K discussed in Sec. III B. The RDFs from both the bulk and interface at 200 K are shown in Fig. 9, with both r₁(ρ) and r₂(ρ) annotated.

FIG. 9.

Structural criteria to determine the possibility of BUMPer to detect structural anomalies for different systems. Presented RDFs are calculated from the CG COM–COM pair distributions where the pair distances at the first maximum and second maximum are marked as r₁(ρ) and r₂(ρ) (dotted lines), respectively. (a) Bulk BUMPer at 200 K. (b) Ice/water interfacial BUMPer at 200 K.

Even though both RDFs generate comparable contours, we note that the RDF of the interface system is more structured at distances beyond 4 Å, as exhibited by the sharper peaks. The general profile presented in bulk supercooled water is much softer even under supercooled conditions, confirming the design principles of our model. For the bulk, the distance at the first maximum is 2.75 Å, and the second maximum distance is 4.65 Å, giving a ratio of 0.591. The ratio from the interface system, 0.6071, also satisfies the suggested criterion despite a slightly larger value than the bulk case with $r_1^{\text{interface}}\left(\rho \right)=2.75$ Å and $r_2^{\text{interface}}\left(\rho \right)=4.53$ Å.

4. Low-temperature behavior extracted from CG PMF

Alternatively, other criteria based on the profile and shape parameters of potentials have been suggested to indirectly determine the ability of the CG model to predict the anomalous behavior of water.^153–156 Hence, in this section, we aim to apply these semiempirical criteria to BUMPer to predict its ability to display anomalous behavior. These criteria were initially designed from the discontinuous shouldered well (DSW)¹⁴⁴ potential U_DSW(R) that is defined as

$$U_{\text{DSW}}\left(R\right)=\left\{\begin{array}{cc}\infty \hfill & \text{for\hspace{0.17em}}R<a,\hfill \\ U_R\hfill & \text{for\hspace{0.17em}}a\le R<R_R,\hfill \\ -U_A\hfill & \text{for\hspace{0.17em}}{R}_R\le R<R_A,\hfill \\ 0\hfill & \text{for\hspace{0.17em}}{R}_A\le R,\hfill \end{array}\right.$$ (12)

where U_R and U_A refer to the energies of the short-range repulsive shoulder and long-range attractive well in a two-scale potential. The distance a denotes the repulsive “inner core” regime, and R_R and R_A correspond to the repulsive average radius and the distance at the attractive minimum. It has been shown that the DSW potential can exhibit dynamic and structural anomalies correctly but does not show thermodynamic anomalies.¹⁴⁴ Recently, de Oliveira et al.⁹⁹ resolved this limitation by employing a continuous generalization of such a shouldered potential from the work of Franzese.¹⁵⁶ The so-called CSW as defined in the following equation correctly recapitulates the hierarchical anomalies that are characterized by water:⁹⁹

$$U_{\text{CSW}}\left(R\right)=\frac{U_R}{1+\text{exp}\left(\frac{\mathrm{\Delta }\left(R-R_R\right)}{a}\right)}-U_A\cdot \text{exp}\left[-\frac{{\left(R-R_A\right)}^2}{2{\delta }_A^2}\right]+{\left(\frac{a}{R}\right)}^{24}.$$ (13)

Both CSW and DSW potentials are guaranteed to have a global minimum of −U_A. From the variables in Eq. (12), the widths of the repulsive and attractive wells are defined as w_R = R_R − a and w_A = R_A − R_R.

As witnessed from the hierarchical anomalies exhibited by a wide range of interaction parameters, the following empirical criterion has been suggested¹⁵⁵ and then employed^99,156,157 to predict if the pairwise CG interaction is capable of emulating the anomalies of water,

$$-2\le \frac{1}{U_A{V}_{SC}}{\int }_a^{\infty }U\left(\mathit{R}\right)\cdot d\overrightarrow{\mathbf{R}}\le 1.$$ (14)

Equation (14) is derived from the DSW model using the modified van der Waals equation of state, which is a mean-field approach of describing phase transitions in terms of potential parameters,¹⁵⁵ and later corroborated in the case of continuous shouldered potentials as well.¹⁵⁶ For simplicity, we define the “criterion integral” C(R) ≔ $\frac{1}{U_A{V}_{SC}}{\int }_a^{\infty }U\left(\mathit{R}\right)\cdot d\overrightarrow{\mathbf{R}}$ in Eq. (14), where $V_{SC}=\frac{2\pi }{3}{R}_R^3$ denotes the soft-core volume within the repulsive regime.

These criteria have a clear limitation: Eq. (14) is designed for shouldered well potentials, and there are no other proposed criteria for the double-well potential. Nevertheless, we now apply these criteria to the BUMPer interaction based on the following observation from a prior bottom-up CG water model derived using relative entropy minimization. As the density changed under four different conditions, Chaimovich and Shell observed a continuous transformation between a double-well shape to a shouldered form under the same atomistic force field.¹⁵⁸ They further claimed that the same basic functionality with two characteristic scales remains in either the double-well form or the ramp with a single-well form as long as the pair interaction has three inflection points and an attractive well, indicating that Eq. (14) can be applied to BUMPer.

Yet, even for double-well interactions, the effective short-range repulsive shoulder width w_R is required to calculate the criterion integral shown in Eq. (14). We approached this problem by extracting the minimum and maximum possible values of w_R from our BUMPer interactions that correspond to short-range shoulder and ramp potentials, respectively. Then, we calculated the criterion integral for both limiting cases to see if the condition is satisfied within this range of w_R. First, approximating the short-range region as a shouldered interaction, or “shoulder scheme,” assumes that the w_R corresponds to the minimum width of the short-range well in a double-well form. This scheme is based on the continuous transformation reported by Shell that the core-softened interactions will become a shouldered well interaction as density increases.¹⁵⁸ The other limiting situation is where the short-range regime and subsequent barrier between the two wells merge into a larger ramp, or “ramp scheme.” In this condition, the core-softened interaction becomes a repulsive ramp with maximum width. Therefore, by considering these two limiting cases schematically illustrated in Fig. 10, we can indirectly examine all possible double-well or shouldered-well interaction forms derived from BUMPer.

FIG. 10.

Schematic diagram of extracting minimum and maximum in the possible range of the repulsive width w_R from the given BUMPer interaction. (a) Shoulder scheme: the possibly minimum w_R value corresponds to the width of the first well. (b) Ramp scheme: the possibly maximum w_R value corresponds to the region that encompasses the well and barrier region.

In other words, to mitigate transferability issues at different density conditions and maximize the efficiency of parameterization, we considered both minimum and maximum w_R values based on the interaction parameterized at a single state point: the shoulder scheme should give the minimum w_R value, while the maximum w_R value should be obtained by the ramp scheme. While w_R for the shoulder scheme can be obtained straightforwardly as mentioned earlier, a constant repulsive ramp needs to be carefully designed. In order to maintain the same level of accuracy in reproducing the many-body CG PMF from this modified ramp-like interaction, we designed the ramp interaction to conserve the overall energy from the CG potential during the ramp scheme. In this way, even though the constant ramp interaction U_R cancels out the attractive and repulsive fluctuations, the overall CG energy will be still identical to that of the original BUMPer system via the energy-conserving mapping, as given by the following equation:

$${⟨U\left(R_1\le R\le R_2\right)⟩}_{\text{CG}}={⟨U_R\left(R_1\le R\le R_2\right)⟩}_{\text{CG}},$$ (15a)

$$4\pi \rho {\int }_{R_1}^{R_2}\left[U\left(R\right)-U_R\right]\cdot R^{2}g\left(R\right)\cdot dR=0,$$ (15b)

where ⟨·⟩_CG denotes the CG ensemble average value. Using the structural average on the basis of pairwise distance, Eq. (15a) can be rewritten as an integral formalism to give Eq. (15b), where R₁ and R₂ are the minimum and maximum distances that satisfy U(R₁) = U(R₂) = U_R, respectively, ρ denotes the (number) density of the system, and g(R) is pair correlation function. We notice that Eq. (15) is similar to the Maxwell construction in thermodynamics,¹⁵⁹ but in this case, the integration is performed over the radial domain with weights based on the pair distribution g(R). In particular, Eq. (15) determines the optimal ramp interaction that accounts for a net cancellation between attraction and repulsion due to a double-well form. In a practical manner, we determine U_R by numerically solving the integral equation [Eq. (15b)] to obtain the effective w_R value for the ramp scheme: w_R = R₂ − R₁.

We applied the energy-conserving mapping to four different BUMPer models derived from different atomistic force fields for water to test the efficacy of the method. Figure 11 shows the original BUMPer and mapped ramp interactions. We observe that the parameterized U_R values for each of the four CG models are positive (repulsive ramp), satisfying the design of the Stell–Hemmer-based model. At first glance, this is surprising since Eq. (15) does not directly imply U_R > 0. Nevertheless, this behavior can be understood from the observation noted in Ref. 155 that a repulsive ramp is necessary to reach the balance between the attractive and repulsive part of the interactions to demonstrate anomalous behavior in water. Similarly, one could obtain U_R for the shoulder scheme as well using the energy-conserving mapping condition, but for the shoulder scheme, it is not necessary to solve this mapping since C(R) is not dependent on U_R values. Finally, using the fitted parameters from both schemes, we computed the C(R) values for the four BUMPer CG PMFs at 300 K. Table II below shows the computed values using numerical integration. We note that the numerical integration was performed up to the pair distance where the long-range attraction decays to zero,

$$\begin{array}{ccc}\hfill C\left(R\right)& \approx \hfill & \frac{1}{U_A{V}_{SC}}{\int }_a^{R_A}U\left(\mathit{R}\right)\cdot d\overrightarrow{\mathbf{R}}\hfill \\ \hfill & =\hfill & \frac{3}{U_{A}2\pi {\left(a+w_R\right)}^3}{\int }_a^{R_A}4\pi R^{2}U\left(R\right)\cdot dR.\hfill \end{array}$$ (16)

FIG. 11.

Mapping the short-range ramp repulsion from the BUMPer interactions by conserving the overall CG energy. The ramp interaction (blue line) is determined by effectively conserving the overall CG free energy by compensating the attractive and repulsive fluctuations (red dashed line). Then, w_R is obtained from the effective ramp with the well model (solid lines). Four BUMPer CG models are considered that are parameterized from (a) SPC/E, (b) SPC/Fw, (c) TIP4P/2005, and (d) TIP4P/ice force fields.

TABLE II.

Fitted parameters from the BUMPer CG PMFs parameterized by four different atomistic force fields at 300 K. Parameters a, w_A, and U_A are obtained by matching the DSW and CSW potential profile. We consider two limiting cases of the w_R parameter using the shoulder scheme (gives minimum) and ramp scheme (gives maximum) from the given BUMPer interactions. From the parameters, we numerically computed the criterion integral C(R).

				Shoulder scheme		Ramp scheme
System	a (Å)	wA (Å)	UA (kcal/mol)	wR (Å)	C(R)	wR (Å)	C(R)
SPC/E	2.694	2.640	0.193	0.84	−0.871	1.194	−0.879
SPC/Fw	2.700	2.760	0.218	0.84	−0.734	1.228	−0.739
TIP4P/2005	2.664	2.520	0.189	0.96	−0.348	1.236	−0.283
TIP4P/ice	2.708	2.480	0.197	0.88	−0.578	1.174	−0.457

Table II shows that the C(R) values for the four different BUMPer CG models are between −2 and 1, indicating their possibility to display hierarchical anomalies. We observe that C(R) values computed at both minimum and maximum widths w_R satisfy the criterion and, thus, we also argue that intermediate w_R values would also fall in the region where Eq. (14) is still valid. However, we note that the parameters a, w_R, w_A, and U_A in Table II are obtained at 300 K. Given the temperature-dependent nature of the CG PMF, an extrapolation scheme to temperatures lower than 300 K is needed to provide accurate structural predictions at low temperatures.

5. Low-temperature behavior extracted from temperature transferred CG PMF

In this section, we elucidate the low-temperature behavior of BUMPer by combining the ansatz used in Sec. III A with the temperature transferability addressed in Subsection III C 1. In other words, we apply Eq. (6) to obtain CG interactions and the corresponding C(R) values at temperatures lower than 300 K. Free energy decomposition-based extrapolation from liquid phase calculations is suitable for predicting CG interactions of supercooled water, as the system still contains liquid-like character, a key assumption of free energy decomposition, whereas crystallized ice at the same temperature likely exhibits a different CG interaction profile.

We note that developing a fully transferable BUMPer model over thermodynamic state points will require a systematic understanding of the changes in CG PMF under different state points, which is beyond the scope of this paper. However, an approximation using the Helmholtz free energy under constant volume conditions should be reasonable as the FG pressure only changes by a small amount in this temperature range, and the criterion integral evaluated from any possible shape of the BUMPer interactions should fall between these two limiting cases (Fig. 10). We employed BUMPer from the TIP4P/ice model, which is consistent with Sec. III B on the ice/water interface. We chose temperatures between 280 K and 360 K to obtain the pairwise entropy and energy functions and extrapolated the potential down to 200 K. The temperature range of 200 K–280 K coincides with the previously reported anomalous region.⁶⁰ From the extrapolated CG interactions, we applied the shoulder and ramp schemes to obtain the minimum and maximum possible w_R values.

As demonstrated in Fig. 12(a), we revalidate the linear dependence of the CG PMF with respect to temperature. We observe that the fitted BUMPer CG PMFs will have deeper attractive wells as temperature decreases from 360 K, since low-temperature water is more structured. Based on the fitted CG PMFs, we mapped the corresponding ramp interactions using Eq. (15). Using two different repulsive widths w_R, we calculated the C(R) values at low temperatures, as shown in Fig. 12(b). The final fitted parameters for the CSW form are presented in Table III with the corresponding C(R) values.

FIG. 12.

Assessing the criteria for low-temperature anomaly by constructing a temperature transferable BUMPer. (a) BUMPer CG potentials at different temperatures from 200 K (red) to 360 K (blue). From the parameterized CG interactions at 300 K–360 K, we extrapolated the low-temperature potentials down to 200 K. (b) Calculated C(R) values at directly parameterized temperatures (red) and extrapolated regimes (blue) using both ramp (circle point) and shoulder scheme (diamond point). The possible ranges of C(R) values by changing w_R are filled in between two schemes, satisfying Eq. (14).

TABLE III.

Fitted parameters from the BUMPer CG PMFs at different temperatures. Parameters a, w_A, and U_A are obtained by matching the DSW and CSW potential profile. We consider two limiting cases of the w_R parameter using the shoulder scheme (gives minimum) and ramp scheme (gives maximum) from the given BUMPer interactions. From the parameters, we numerically computed the criterion integral C(R).

				Shoulder scheme		Ramp scheme
Temperature (K)	a (Å)	wA (Å)	UA (kcal/mol)	wR (Å)	C(R)	wR (Å)	C(R)
360	2.776	2.520	0.184	0.80	−0.532	1.134	−0.407
340	2.746	2.520	0.187	0.80	−0.497	1.146	−0.376
320	2.710	2.480	0.190	0.88	−0.525	1.170	−0.416
300	2.708	2.480	0.197	0.88	−0.578	1.174	−0.457
280	2.674	2.480	0.200	0.88	−0.631	1.198	−0.488
260	2.652	2.480	0.203	0.92	−0.659	1.224	−0.525
240	2.638	2.480	0.206	0.92	−0.713	1.234	−0.564
220	2.624	2.480	0.209	0.92	−0.785	1.246	−0.617
200	2.608	2.480	0.211	0.96	−0.834	1.258	−0.669

As listed in Table III, we find that all computed C(R) values at temperatures ranging from 200 K to 280 K (lower than the freezing point) are in the criterion range suggestive of hierarchical anomalies. Similar to the results at 300 K, Eq. (14) is satisfied using both shoulder and ramp schemes, as depicted in Fig. 12(c), indicating that any possible values of w_R from the BUMPer interactions are expected to exhibit anomalous behavior.

6. Examination of low-temperature behavior in other CG models

We further corroborate the fidelity of the proposed protocol, i.e., the shoulder and ramp schemes, by applying our analysis to other bottom-up CG models. As demonstrated in Fig. 6 and Sec. III B 3, the SP-FM CG model is not capable of capturing the structural correlations at the ice/water interface, and we expect that this model will be unable to capture low-temperature anomalies. To test this hypothesis, we extrapolated the parameterized interactions (from 280 K to 360 K) down to 200 K, as shown in Fig. 13(a). Figure 13(b) supports our claim in which the C(R) criterion from Eq. (14) is no longer satisfied at temperatures lower than the freezing point.

FIG. 13.

Assessment of low-temperature anomaly extracted from the SP-FM CG model. (a) Temperature-dependence of the CG PMFs from 200 K (red) to 360 K (blue). From the parameterized CG interactions at 300 K–360 K, we extrapolated the low-temperature potentials down to 200 K. (b) Calculated C(R) values at directly parameterized temperatures (red) and extrapolated regimes (blue) using both ramp (circle point) and shoulder scheme (diamond point), where the possible C(R) values with different w_R values are located between two schemes. Gray area denotes the region where Eq. (14) is violated.

The mW-BUMPer model, which lacks temperature transferability, also does not satisfy this criterion, since the interaction form depicted in Fig. 4(b) does not follow the double-well or shouldered-well shape. Hence, these violations from other CG models indicate that low-temperature anomalies are exhibited if and only if the CG models can adequately capture structural correlations and address temperature transferability correctly.

Taken together, the two different criteria based on the structural correlations and CG PMFs suggest that BUMPer encodes the anomalies of water at low temperatures, whereas some CG potentials do not exhibit such behavior. Determination of the criterion integral from the shoulder and ramp schemes further confirms the previous observation from Chaimovich and Shell in a quantitative way,¹⁵⁸ as it is also in line with a statistical mechanical understanding of the shouldered well potentials in the literature. Nonetheless, we seek to provide a more direct approach of demonstrating the low-temperature anomalies of water by designing BUMPer that is transferable to different densities and temperatures, which will be the topic of a subsequent paper of this series. One possible direction is to evaluate the excess entropy from the CG system and then to understand the corresponding structural anomalies.^160,161 However, this requires a comprehensive understanding of the entropy representability relationship between the FG and CG systems.^36,37 This work is underway.

IV. CONCLUSIONS AND FUTURE DIRECTIONS

In this second paper of the series, we analyzed the low-temperature behavior and anomalies of the BUMPer CG model for water. Given that bottom-up CG interactions are essentially free energy quantities, a direct application of the developed BUMPer (parameterized from room temperature) to much lower temperatures is not possible without modification. Using a free energy decomposition scheme, we accounted for an explicit temperature-dependent (entropic) contribution to the CG interaction, resulting in a transferable water model that spans all temperature ranges accessible to the liquid phase at standard pressure. Hence, full temperature transferability in the liquid phase is addressed by BUMPer.

Direct parameterization using low-temperature reference data is possible as well. We constructed an ice/water interface near the freezing point as a prototypical system for demonstrating the growth of ice and derived CG models at this state point. We report, for the first time, that a bottom-up CG model described by only pairwise interactions is capable of simulating ice crystal growth. We attribute this success to the projection of many-body interactions into the pairwise BUMPer potential after analysis of structural correlations in the RDF and LSI.

To delve into the supercooled regime much lower than the freezing point, we assessed the possibility of BUMPer to recapitulate structural and dynamical anomalies reported in supercooled water. As direct CG simulation under supercooled conditions are not feasible due to the transferability problem, we utilized two different criteria proposed in the literature by assuming that the pairwise interaction profile can be regarded as short-range repulsion with long-range attraction. By considering both continuous shouldered well-shape and ramped well-shape interaction forms, we quantitatively demonstrated that BUMPer will follow the generalized empirical criteria under low-temperature conditions, while some CG models seem to violate these criteria.

The findings presented in this work provide a systematic way of constructing bottom-up CG water models under supercooled conditions. At the same time, our analysis reveals several practical problems that can be pursued as future research directions. Although ice growth at the interface is faithfully captured by BUMPer, direct simulation of nucleation from homogeneous bulk conditions should be explored. Given the large activation energy, brute-force simulation of homogeneous nucleation using only pairwise potentials are still challenging, and we are currently working to address this issue by combining BUMPer with enhanced sampling techniques, such as metadynamics.^162,163 To precisely address structural correlations at any state point, a BUMPer potential that accounts for changes in pressure, temperature, and chemical potential should be developed. To this end, going beyond the temperature transferability shown in this work, we seek to provide a CG model of water with an explicit dependence on key state variables in the next paper of this series. Nonetheless, our findings so far point to a potential use of BUMPer not only at room temperature but also at supercooled temperatures and in heterogeneous systems such as solvated biomolecules at the CG level. In these cases, we anticipate that the many-body projection theory can provide an accurate bottom-up CG model and serve as a building block for constructing transferable models while maintaining the low computational cost, which is a key element of any CG approach.

APPENDIX A: SIMPLE PAIRWISE CG MODEL FOR ICE/ WATER INTERFACE

Figure 14 depicts the SP-FM CG interaction that is obtained from the ice/water interface at 249 K. When compared to the bulk liquid system at 249 K that is extrapolated using Eq. (6), we discover that both CG interactions give almost exact short-range behavior, while some differences are noted at long distances (>4 Å). These differences are ascribed to the long-range ordering that may be important in the interfacial system.

FIG. 14.

SP-FM CG models for the interface (blue solid line) and bulk (blue dashed line) at 249 K.

APPENDIX B: INHERENT LOCAL STRUCTURE INDEX

Figure 15 summarizes the computational protocol to generate the inherent MD trajectory for computing the inherent local structure index. We first collected ambient MD configurations every 1 ps during the CG simulations for 5 ns. Then, we performed energy minimization for each configuration using the corresponding CG interactions via conjugate gradient minimization for up to 10 000 steps, followed by re-optimization with the steepest descent algorithm to generate an energy-minimized snapshot. Finally, we concatenated each minimized inherent CG configuration to obtain a single CG trajectory for the LSI analysis.

FIG. 15.

Detailed protocol used to generate the inherent MD trajectory for this work. From the ambient (regular) MD trajectory at non-zero temperatures (T = 280 K–360 K in this work), we separated the overall trajectory into various sub-trajectories by dumping the sub-trajectory as the same frequency used to dump restart files. For each sub-trajectory, we performed LAMMPS rerun^164–166 to minimize the energy of the CG configurations at the given sub-trajectory followed by the conjugate gradient and steepest descent minimization algorithms. After finishing the minimization, we combined the energy-minimized sub-trajectories into a single trajectory, which is now “inherent” MD trajectories corresponding to the energy-minimized configurations at T = 0 K.

DATA AVAILABILITY

The BUMPer CG model, presented and used within this study, is openly available in Github.⁹⁸ Additional data and analysis scripts supporting the findings of this study are available from the corresponding author upon request.