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Experimental studies have reported that zinc oxide improves the durability of glasses used for nuclear waste immobilization. Here, molecular dynamics simulations are used to predict the atomic structures of sodium silicate glass with and without zinc. A simulated melt-quench procedure is used to generate glass structures. Pair distribution functions, ring size distributions, and alkali clustering are then examined. This allows insights into the structural role of zinc oxide within the glass and helps distinguish between its reported functions as a network former and a network modifier. Changes in the sodium ion distribution and clustering behavior within the glass are observed, due to zinc oxide addition. This affects the local and intermediate-range structure of the glass and provides a possible explanation for enhanced durability.