Numerical determination of the exponents controlling the relationship between time, length, and temperature in glass-forming liquids

There is a certain consensus that the very fast growth of the relaxation time tau occurring in glass-forming liquids on lowering the temperature must be due to the thermally activated rearrangement of correlated regions of growing size. Even though measuring the size of these regions has defied scientists for a while, there is indeed recent evidence of a growing correlation length xi in glass formers. If we use Arrhenius law and make the mild assumption that the free-energy barrier to rearrangement scales as some power Sigma of the size of the correlated regions, we obtain a relationship between time and length, T log tau similar to xi(Sigma). According to both the Adam-Gibbs and the random first order theory the correlation length grows as xi similar to(T-T-k)(-1/(d-theta)), even though the two theories disagree on the value of theta. Therefore, the super-Arrhenius growth of the relaxation time with the temperature is regulated by the two exponents Sigma and theta through the relationship T log tau similar to(T-T-k)(-Sigma/(d-theta)). Despite a few theoretical speculations, up to now there has been no experimental determination of these two exponents. Here we measure them numerically in a model glass former, finding Sigma=1 and theta=2. Surprisingly, even though the values we found disagree with most previous theoretical suggestions, they give back the well-known VFT law for the relaxation time, T log tau similar to(T-T-k)(-1).