This paper introduces a novel approach for topology optimization of differentiable microstructures, which are microstructures that can be continuously parameterized and exhibit smooth variations in both geometry and mechanical properties. The method uses a heat equation to encode the non-uniform expansion of an initial microstructure with a low volume fraction, generating a series of microstructures with progressively increasing volume fractions. The key idea is to use a continuous parameterization of microstructures, allowing for smooth transitions between different microstructures and eliminating the need for quantization or multi-material optimization. The method is validated by comparing the results with alternative approaches and analyzing the geometric features and differentiability of the optimized microstructures. The results show that the proposed method achieves a bulk modulus that is much closer to the theoretical limit compared to alternative methods. The method is also shown to be effective in two-scale topology optimization, where the continuous parameterization allows for a more accurate representation of the microstructure's mechanical properties. The paper concludes that the proposed method provides a promising approach for the design of functionally graded microstructures with optimal mechanical properties.This paper introduces a novel approach for topology optimization of differentiable microstructures, which are microstructures that can be continuously parameterized and exhibit smooth variations in both geometry and mechanical properties. The method uses a heat equation to encode the non-uniform expansion of an initial microstructure with a low volume fraction, generating a series of microstructures with progressively increasing volume fractions. The key idea is to use a continuous parameterization of microstructures, allowing for smooth transitions between different microstructures and eliminating the need for quantization or multi-material optimization. The method is validated by comparing the results with alternative approaches and analyzing the geometric features and differentiability of the optimized microstructures. The results show that the proposed method achieves a bulk modulus that is much closer to the theoretical limit compared to alternative methods. The method is also shown to be effective in two-scale topology optimization, where the continuous parameterization allows for a more accurate representation of the microstructure's mechanical properties. The paper concludes that the proposed method provides a promising approach for the design of functionally graded microstructures with optimal mechanical properties.