Formose toy model

C1 permeability

Experimentally, a 'flat line' is observed for $C1$ exchange at 8M vs 0.8M. Below, different permeability regimes are illustrated for this scenario, using the above model without reaction. As expected, the toy model gives no discernable growth when $P_{C1}/P_{H_2O} \approx v_{H2O}/v_{C1}$. Flatness is obtained here for $P_{C1}/P_{H_2O}=0.6$. However, when $P_{C1}\gg P_{H_2O}$, the slight initial volume divergence (due to exchange of C1) may occur on a timescale that is not experimentally accessible, so that volume curves would appear flat.

Effect of distance and molar volumes for larger species

Experimentally, we observe that for larger species ($C_n, n \geq 3$), in the absence of the formose reaction, volume time courses appear as linear lines and the intersection point position changes with distance and sugar size (their molar volume). We illustrate these trends with the model below.

We first test the effect of distance for a difference of $C_2$ concentration for different droplet spacings.

We now test the effect of molar volume for larger species, which permeability is very poor. We extrapolate $v_{Cn}=0.029+0.012(n-1) L.mol^{-1}$.

We observe that the effect of molar volume is very small.

Formose reaction in droplets

First, let us check that when the permeabilities are set to 0, the kinetics of the toy model are recovered.

We now explore the model with simultaneous exchange and the formose reaction. Note that, unlike in the sugar exchange experiments, the system here contains other species such as methanol (~0.15 M), TMG (1 M), CaCl2 (0.02M).

TMG occupies a considerable volume and is a strong chaotrope. Therefore, the hydrogen bonding in the aqueous droplet phase is destabilized and it becomes energetically less favorable for $H2O$ and hydrophilic species to reside in the aqueous droplet phase.

Since permeability is proportional to the partition constant, our null hypothesis should correspondingly be that H2O, C1 and C2 become more permeable. Below, we retain their original ratios but vary their absolute magnitude to see at what point they produce exchange comparable to what is observed for 20 $\mu m$ separation.

We now examine the dynamics of species in each of the coupled droplets.

Phase diagram

Let us now explore more systematically various regimes afforded by varying the timescales of the reaction and transport.

Varying $\tau_{C1}, \tau_{cat}$

To obtain a characteristic exchange time of C2, we consider a droplet with C2 at concentration $c$ coupled to a droplet without C2. The flux is $P_{C2} A_{red} x_{C2}$ in $mol.min^{-1}$ with $x_{C2}=n_{C2}/n_{tot} \approx n_{C2}/n_{H2O}$ and transports a characteristic amount of C2 of $n_{C2}$ in moles. In fine, $\tau_{C2} \approx c_{H2O} V / (P_{C2} A_{red})\approx c_{H2O}D/P_{C2}$ where $D$ is the droplet diameter.