In the context of a circular building material economy, the complexity of resource flows and the variability of material composition pose significant challenges. This study demonstrates how Large Language Models (LLMs) can advance material design by adopting a Knowledge-Driven Design (KDD) approach that outperforms traditional Data-Driven Design (DDD) methods. Our focus is on designing alkali-activated concrete (AAC) mix designs, an environmentally friendly alternative to conventional Portland cement-based concrete. GPT-3.5 Turbo and GPT-4 Turbo enable using fuzzy design knowledge as previously untapped input data modality. A key aspect of our research is to improve the performance of the LLMs in post-training. We use strategies such as refining prompt context, extending test time, and including a verifier.The study's systematic benchmarks are based on 240 AAC formulations extracted from the literature. The target was on achieving maximum compressive strength through an adaptive design approach over multiple development cycles. We compare these results to the traditional DDD baseline methods. KDD outperforms conventional methods by providing robust initial predictions and demonstrating effective adaptability informed by laboratory validation data, culminating in the development of high-quality AAC formulations. These results provide valuable insight into the capabilities of chat-based LLMs in managing complex material formulations, which are particularly beneficial in situations where traditional DDD is impractical due to data collection issues. With natural language as the basis the KDD is intuitively accessible to domain experts. The methodology and results of this study have significant implications for the field of materials science, particularly in the context of a circular economy, and pave the way for innovative applications of LLMs in various scientific fields.