Good question, I'm not an expert but I would like to share my view.

In my opinion, in a longitudinal study both techniques (repeated measures ANOVA and LMM) are equal when you don't have contextual differences within your subjects (e.g., your participants come from different hospitals, universities, whatever). LMM allow you to include random effects in your model, which might affect your results but you are not really interested in them. In general, LMM are more flexible when you construct the model.

However, your design looks like a basic experiment, in which normally repeated measures anova is enough (as long as statistical assumptions are checked and met).

Do you think your experiment needs to model random effects?

Hope this helps!

Thank you, for your response true, I don't need to model random effects. I just need to see independent group condition effect on DV's between and within group.

So to find between group effects and within group effect than mixed effect will be appropriate?

WIth 4 measurements during day 1 and only one later, is your interest in variation during day 1, or between day 1 and day 6? What are the correlations between these like? What is your n? How are they distributed?

I don't understand what you mean "I don't need to model random effects." Do you mean you just to include, say, a random slope but don't expect it to be a function of any other variables?