Sunflowers are susceptible to multiple pests and diseases, such as Sunflower Beetles, Cutworms and Sunflower Borers, powdery mildew, and rust. But by far, sunflowers are most susceptible to Sclerotinia, also known as "white mold". Without being able to view an actual sample, from the photograph you sent, we suspect your flowers to be infected with Sclerotinia head rot. This is an air borne disease.
You can read more about Sclerotinia and other possible causes of your sunflower symptoms at the following websites.
In addition, we suggest you take your plant to your local extension office in order to obtain a firm diagnosis.
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