In order to provide insights into the misfolding mechanism and the subsequent aggregate formation which cause what are known as the neurodegenerative polyglutamine diseases, we have simulated a 10-residue polyglutamine (poly-Q) chain in vacuum and in solvent by multicanonical method, which enabled us to study the system in a wide temperature range and discuss thermodynamic properties. It is understood that the system in vacuum shows two phase transitions, first of them occur at high temperature that is the well-known helix-coil transition and the second one is a solid-solid transition. However, the poly-Q chain in solvent is in a random coil state at higher temperatures, goes through a conformational change at T = 200 K and assumes predominantly a mixture of anti-parallel β-sheet and α-helix structures at low temperatures. One-residue glutamine dipeptide is also simulated and low-energy stable conformations are identified.