## 1. Introduction

[2] It is well known that the phase state of an atmospheric aerosol particle plays an important role in governing its composition, water content, size, and direct radiative forcing [*Wexler and Seinfeld*, 1991; *Seinfeld and Pandis*, 1997; *Tang*, 1997; *Martin et al.*, 2004]. At thermodynamic equilibrium, a particle made up of a single salt will be solid if the ambient relative humidity (RH) is below its deliquescence RH (DRH) and will be completely dissolved (liquid) at RH ≥ DRH. Once completely deliquesced, the solution droplet will continue to grow by water vapor condensation as RH further increases. Conversely, as RH decreases, water vapor will evaporate and the solution droplet will continue to shrink. Numerous laboratory experiments have also shown that a completely deliquesced particle can remain in a metastable liquid solution that is supersaturated with the salt until crystallization finally takes place at relative humidities that are much below the DRH [*Tang et al.*, 1995]. While the DRH for a given salt depends only on temperature, the crystallization RH (CRH) is a complex function of homogeneous nucleation of the solid phase, and is sensitive to solid impurities in the solution [*Lightstone et al.*, 2000; *Onasch et al.*, 2000].

[3] Tropospheric particles typically contain a mixture of electrolytes of ammonium, sodium, calcium, sulfate, nitrate, chloride, etc. Hygroscopic growth of such mixed-salt particles also behaves in a similar manner as described above except that it exhibits a complex multistage deliquescence behavior instead of a single deliquescence point. This is illustrated in Figure 1 with the example of an equimolar mixture of Na_{2}SO_{4} and NaCl [*Tang*, 1997]. As single salts, NaCl and Na_{2}SO_{4} deliquesce at ∼75.3% and ∼87% RH, respectively, at 298.15 K. However, as a mixture, the solid particle partially deliquesces first at ∼74.2% RH when NaCl is completely dissolved while some solid Na_{2}SO_{4} remains in equilibrium with the aqueous phase. As RH further increases, more Na_{2}SO_{4} continues to dissolve into the aqueous phase until it is completely deliquesced at ∼80% RH. As the RH is now decreased, the liquid particle remains in a metastable state until ∼52.5% RH when it spontaneously crystallizes. The first deliquescence point is called the mutual deliquescence RH (MDRH) and the point when the particle is completely dissolved is called the complete deliquescence RH (CDRH). The point where a completely deliquesced particle finally crystallizes during efflorescence (evaporation) is called the mutual crystallization RH (MCRH).

[4] At a given temperature, it has been shown that MDRH of a salt mixture corresponds to the eutonic point of the mixture, and is lower than the minimum DRH of each salt. Moreover, it depends only on which salts are present in the dry state, but not their relative amounts [*Wexler and Seinfeld*, 1991; *Tang and Munkelwitz*, 1993; *Potukuchi and Wexler*, 1995a, 1995b]. On the other hand, each subsequent partial deliquescence points during the multistage growth as well as the CDRH depend on the relative amounts of the salts present. For example, *Tang et al.* [1978] experimentally showed that particles having different initial compositions in wt% of NaCl and KCl, respectively, of 80-20, 64-36, and 20-80 exhibited the first deliquescence point (i.e., MDRH) at 73.8% RH, which is lower than the respective individual DRHs of 75.3% and 84.3%. However, the CDRH of each mixture was different, and ranged between 73.8% and 84.3% RH depending on the relative amounts of each salt.

### 1.1 Current Thermodynamic Equilibrium Models

[5] Accurately predicting the multistage deliquescence points and the associated solid-liquid partitioning is a challenging task. In the recent years, numerous thermodynamic models have been developed that attempt to solve this problem using different numerical techniques – SCAPE2 [*Kim et al.*, 1993a, 1993b; *Kim and Seinfeld*, 1995; *Meng et al.*, 1995], ISORROPIA [*Nenes et al.*, 1998], EQUISOLV II [*Jacobson et al.*, 1996; *Jacobson*, 1999], GFEMN [*Ansari and Pandis*, 1999b], and AIM [*Clegg et al.*, 1998a, 1998b; *Wexler and Clegg*, 2002].

[6] SCAPE2 and ISORROPIA classify a given multicomponent problem into one of the several sub-domains based on the relative composition where only certain components/reactions are assumed to occur. The relevant equilibrium equations in a chosen sub-domain are then solved with the bisection method in an iterative manner. However, SCAPE2 does not predict MDRH as a function of composition. Instead, it assumes that the particle first deliquesces at the lowest DRH of all the individual salts present in a given mixture. While ISORROPIA predicts MDRH as a function of composition and temperature, it assumes a “mutual deliquescence region” between MDRH and the lowest DRH (denoted as RH_{wet}) of all the individual salts present in the mixture. Moreover, for any RH between MDRH and RH_{wet}, the aerosol water content and dissolution of different salts in ISORROPIA are approximated using a weighted mean of the “dry” state (MDRH) and the “wet” state (RH_{wet}). Thus, it is evident that the composition and water content of mixed phase systems predicted by SCAPE2 and ISORROPIA will not be at true thermodynamic equilibrium as dictated by the various solid-liquid reactions, and therefore may distort the composition and size of the particle. However, it should be noted that these assumptions and approximations were made to increase the computational efficiency of the models.

[7] On the other hand, EQUISOLV II employs a more rigorous approach to solve the non-linear system of equilibrium equations without any such simplifying assumptions, and therefore adequately predicts the multistage deliquescence behavior of mixed-electrolyte systems. The EQUISOLV II code consists of a numerical scheme that requires the solution of one equilibrium equation at a time either by an analytical technique or by iteration, and all equations are iterated many times until convergence to solve the entire system.

[8] GFEMN and AIM models are rigorous thermodynamic models that directly minimize the total Gibbs free energy of the system using an iterative numerical technique to obtain the equilibrium solution. This approach eliminates many of the assumptions used in SCAPE2 and ISORROPIA such as divided RH and composition domains where only certain reactions are assumed to occur and the various simplifications related to DRH. Also, GFEMN and AIM both use the highly accurate Pitzer-Simonson-Clegg (PSC) [*Pitzer and Simonson*, 1986; *Clegg et al.*, 1992, 1998a, 1998b] module for estimating the multicomponent activity coefficients of aqueous electrolytes, and are therefore regarded as the most accurate thermodynamic models available in the literature. *Ansari and Pandis* [1999a] conducted a comparison of GFEMN with SCAPE2 and ISORROPIA and found that the latter two indeed have difficulty in reproducing the complex multistage deliquescence behavior of mixed-electrolyte systems. A detailed comparative review of EQUISOLV II, SCAPE2, and AIM along with two other models is also available in *Zhang et al.* [2000], and a brief comparison of all the above models and the techniques they use for estimating multicomponent activity coefficients and for predicting solid-liquid partitioning is summarized here in Table 1.

Model Name^{a} | System Solved | Multicomponent Activity Coefficient Method^{b} | MDRH Calculation Method | Solid-Liquid Partitioning Method |
---|---|---|---|---|

- a
SCAPE2 [ *Kim et al.*, 1993a, 1993b;*Kim and Seinfeld*, 1995;*Meng et al.*, 1995]; ISORROPIA [*Nenes et al.*, 1998]; EQUISOLV II [*Jacobson et al.*, 1996;*Jacobson*1999]; GFEMN [*Ansari and Pandis*, 1999b]; AIM [*Clegg et al.*, 1998b;*Wexler and Clegg*, 2002]. - b
Bromley [ *Bromley*, 1973]; KM [*Kusik and Meissner*, 1978]; Pitzer [*Pitzer and Mayorga*, 1973]; PSC [*Pitzer and Simonson*, 1986;*Clegg et al.*, 1992;*Clegg et al.*, 1998a, 1998b;*Wexler and Clegg*, 2002]; MTEM – Multicomponent Taylor Expansion Method [*Zaveri et al.*, 2005]. All models also use the ZSR mixing rule for estimating equilibrium water content [*Zdanovskii*, 1948;*Stokes and Robinson*, 1966]. - c
SCAPE2 assumes that each salt in a multicomponent system deliquesces at its individual (pure state) DRH( *T*). - d
The “wet” state refers to the RH at which the most hygroscopic salt (i.e., with the lowest DRH) is completely dissolved. This weighting scheme assumes that a multicomponent particle is completely dissolved for RH > RH _{wet}. - e
Binary activity coefficients for the electrolytes in the NH _{4}-Na-NO_{3}-SO_{4}-Cl system are temperature dependent, while they are fixed at 298.15 K for the electrolytes in the Ca-Mg-K-SO_{4}NO_{3}-Cl-CO_{3}system. - f
MDRH in these models is not parameterized as a function of particle composition and temperature, but can be computed by rigorously solving the solid-liquid equilibrium problem at varying RH.
| ||||

SCAPE2 | gas-solid-liq: NH_{4}-Na-Ca-Mg-K-NO_{3}-SO_{4}-Cl-CO_{3} | Choice of Bromley, KM, and Pitzer at 298.15 K | Individual salt DRH as a function of temperature^{c} | Iterative bisection method with temperature-dependent equilibrium constants |

ISORROPIA | gas-solid-liq: NH_{4}-Na-NO_{3}-SO_{4}-Cl | Bromley at 298.15 K | Parameterized as a function of composition and temperature | Approximation using a weighted mean of dry and wet solutions^{d} |

EQUISOLV II | gas-solid-liq: NH_{4}-Na-Ca-Mg-K-NO_{3}-SO_{4}-Cl-CO_{3} | Bromley^{e} | N/A^{f} | Analytical equilibrium iteration and mass flux iteration techniques with temperature-dependent equilibrium constants |

GFEMN | gas-solid-liq: NH_{4}-Na-NO_{3}-SO_{4}-Cl | PSC at 298.15 K | N/A^{f} | Iterative Gibbs free energy minimization with temperature-dependent chemical potentials. |

AIM (Model III) | gas-solid-liq: NH_{4}-Na-NO_{3}-SO_{4}-Cl | PSC at 298.15 K | N/A^{f} | Iterative Gibbs free energy minimization at 298.15 K |

MESA (this work) | solid-liq: NH_{4}-Na-Ca-NO_{3}-SO_{4}-Cl | Choice of PSC, MTEM, KM, and Bromley at 298.15 K | Parameterized as a function of composition and temperature | Modified pseudo-transient continuation (PTC) technique with temperature-dependent equilibrium constants |

[9] Numerical techniques based on the direct minimization of Gibbs free energy are highly accurate, but also computationally very expensive and therefore not suitable for inclusion in 3-D chemical transport models [*Ansari and Pandis*, 1999b; *Wexler and Clegg*, 2002]. EQUISOLV II is much more efficient. However, it involves up to three levels of nested iteration loops. For instance, individual equilibrium equations are first iterated (level-3), and when one equation is converged the updated concentrations are used as inputs into subsequent equations (level-2). When the local convergence criterion is met, the water content and activity coefficients are updated, and the level-2 and level-3 iterations are repeated several times (level-1) until the entire system of equations is converged [*Jacobson*, 1999]. While it is possible to solve the second-order equilibrium equations analytically (e.g., NaCl_{(s)} Na_{(aq)}^{+} + Cl_{(aq)}^{−}), level-3 iterations are still required to solve higher order equilibrium reactions (e.g., Na_{2}SO_{4(s)} 2Na_{(aq)}^{+} + SO_{4(aq)}^{2−}) that give rise to third- or higher-degree polynomials. EQUISOLV II requires less than 30 level-1 iterations with the analytical method for typical aerosol composition and ambient RH and temperature conditions; however, the total number of iterations for a given solid-liquid partitioning problem (equal to the product of the number of iterations for each iteration level) is expected to be quite large, especially when third- and higher-order equilibrium reactions are present. The advantage, though, is that the code is generalized to allow the solution to any number and type of equilibrium equations. EQUISOLV II also contains temperature dependence of activity coefficients for the highly sulfate-rich systems containing H^{+}, NH_{4}^{+}, SO_{4}^{2−}, HSO_{4}^{−}, NO_{3}^{−}, and Cl^{−} ions valid at low RH and temperatures found in the upper troposphere and stratosphere. However, temperature dependent parameterizations of binary activity coefficients for all the salts that can form under sulfate-poor conditions (e.g., salts containing Na, Ca, Mg, and K) are either not available or applicable at low RHs, or have not been verified due to lack of experimental data.

### 1.2. Scope

[10] Here, we present a new multicomponent equilibrium solver for aerosols (MESA) based on the pseudo-transient continuation technique for solving solid-liquid equilibria. The algorithm is generally applicable to any number of salts and solid-liquid equilibrium reactions of any order, and involves only one iteration loop. MESA is designed for inclusion in 3-D Eulerian models, and is embedded in a new dynamic gas-aerosol model MOSAIC (Model for Simulating Aerosol Interactions and Chemistry), which will be described elsewhere. The scope of this paper is limited to MESA and the aerosol-phase solid-liquid equilibrium problem. Numerical algorithms for dynamic gas-aerosol partitioning will be described in the future publication on MOSAIC.

[11] The MESA algorithm can be used with different activity coefficient models. *Zaveri et al.* [2005] recently developed a new method, MTEM (Multicomponent Taylor Expansion Method), for estimating activity coefficients of electrolytes in multicomponent atmospheric aerosols. MTEM was applied to aerosol systems containing H^{+}, NH_{4}^{+}, Na^{+}, Ca^{2+}, SO_{4}^{2−}, HSO_{4}^{−}, NO_{3}^{−}, Cl^{−}, and H_{2}O, and was evaluated and contrasted in a stand-alone mode against other multicomponent activity coefficient method such as the comprehensive PSC model and the widely-used *Kusik and Meissner* [1978] (KM) and *Bromley* [1973] mixing rules.

[12] In this paper, we evaluate the performance of MESA using MTEM for several multicomponent systems commonly found in tropospheric aerosols. The Web-based AIM Model III (http://mae.ucdavis.edu/∼wexler/aim.html) is used as a benchmark to evaluate the accuracy of MESA-MTEM. Furthermore, since EQUISOLV II is the only other rigorous equilibrium solver that is efficient enough for use in 3-D models, we use its both non-vectorized and vectorized versions to assess the relative computational speed of MESA-MTEM for the test cases considered in this study.