Abstract The diffusion of water through silicate melts is a key process in volcanic systems. Diffusion controls the growth of the bubbles that drive volcanic eruptions and determines the evolution of the spatial distribution of dissolved water during and after magma mingling, crystal growth, fracturing and fragmentation, and welding of pyroclasts. Accurate models for water diffusion are therefore essential for forward modelling of eruptive behaviour, and for inverse modelling to reconstruct eruptive and post-eruptive history from the spatial distribution of water in eruptive products. Existing models do not include the kinetics of the homogeneous species reaction that interconverts molecular ( H 2 O m ) and hydroxyl ( OH ) water; reaction kinetics are important because final species distribution depends on cooling history. Here we develop a flexible 1D numerical model for diffusion and speciation of water in silicate melts. We validate the model against FTIR transects of the spatial distribution of molecular, hydroxyl, and total water across diffusion-couple experiments of haplogranite composition, run at 800–1200 °C and 5 kbar. We adopt a stepwise approach to analysing and modelling the data. First, we use the analytical Sauer-Freise method to determine the effective diffusivity of total water D H 2 O t as a function of dissolved water concentration C H 2 O t and temperature T for each experiment and find that the dependence of D H 2 O t on C H 2 O t is linear for C H 2 O t ≲ 1.8 wt.% and exponential for C H 2 O t ≳ 1.8 wt.%. Second, we develop a 1D numerical forward model, using the method of lines, to determine a piece-wise function for D H 2 O t C H 2 O t , T that is globally-minimized against the entire experimental dataset. Third, we extend this numerical model to account for speciation of water and determine globally-minimized functions for diffusivity of molecular water D H 2 O m C H 2 O t , T and the equilibrium constant K for the speciation reaction. Our approach includes three key novelties: (1) functions for diffusivities of H 2 O t and H 2 O m , and the speciation reaction, are minimized simultaneously against a large experimental dataset, covering a wide range of water concentration ( 0.25 ≤ C H 2 O t ≤ 7 wt.%) and temperature ( 800 ° C ≤ T ≤ 1200 ° C ), such that the resulting functions are both mutually-consistent and broadly applicable; (2) the minimization allows rigorous and robust analysis of uncertainties such that the accuracy of the functions is quantified; (3) the model can be straightforwardly used to determine functions for diffusivity and speciation for other melt compositions pending suitable diffusion-couple experiments. The modelling approach is suitable for both forward and inverse modelling of diffusion processes in silicate melts; the model is available as a MATLAB script from the electronic supplementary material.

中文翻译：

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流纹质硅酸盐熔体中水的扩散和形态形成的实验验证数值模型

摘要 水通过硅酸盐熔体的扩散是火山系统中的一个关键过程。扩散控制着驱动火山喷发的气泡的生长，并决定了岩浆混合、晶体生长、破裂和破碎以及火山碎屑熔接期间和之后溶解水空间分布的演变。因此，准确的水扩散模型对于喷发行为的正向建模和逆向建模至关重要，以便从喷发产品中水的空间分布重建喷发和喷发后的历史。现有模型不包括分子 (H 2 O m ) 和羟基 ( OH ) 水相互转化的均相反应动力学；反应动力学很重要，因为最终的物种分布取决于冷却历史。在这里，我们开发了一个灵活的一维数值模型，用于硅酸盐熔体中水的扩散和形态形成。我们在 800–1200 °C 和 5 kbar 下运行的单长花岗岩组成的扩散耦合实验中，针对分子、羟基和总水的空间分布的 FTIR 断面验证了模型。我们采用逐步方法来分析和建模数据。首先，我们使用分析 Sauer-Freise 方法来确定总水 DH 2 O t 的有效扩散率作为每个实验的溶解水浓度 CH 2 O t 和温度 T 的函数，并发现 DH 2 O t 对CH 2 O t 对于CH 2 O t ≲ 1.8 wt.% 是线性的，对于CH 2 O t ≳ 1.8 wt.% 是指数型的。其次，我们开发了一个一维数值正向模型，使用线的方法，以确定 DH 2 O t CH 2 O t 的分段函数，T 是针对整个实验数据集全局最小化的。第三，我们扩展这个数值模型来解释水的物种形成，并确定分子水扩散率 DH 2 O m CH 2 O t 、T 和物种形成反应的平衡常数 K 的全局最小函数。我们的方法包括三个关键的创新点：(1) H 2 O t 和 H 2 O m 的扩散系数函数以及物种形成反应，针对大型实验数据集同时最小化，涵盖了广泛的水浓度范围 (0.25 ≤ CH 2 O t ≤ 7 wt.%) 和温度（ 800°C ≤ T ≤ 1200°C ），使得得到的函数既相互一致又具有广泛的适用性；(2) 最小化允许对不确定性进行严格和稳健的分析，从而量化函数的准确性；(3) 该模型可直接用于确定其他熔体成分的扩散系数和形态函数，等待合适的扩散偶实验。该建模方法适用于硅酸盐熔体中扩散过程的正向和逆向建模；该模型可用作电子补充材料中的 MATLAB 脚本。