*2.2. Laboratory-Scale PAUF Experiments*

### 2.2.1. Reagents

Hyperbranched polyethyleneimine (HB-PEI; MW ~25 kDa, 56 wt.% in H2O; Lupasol HF) and modified polyethyleneimine (MOD-PEI; MW ~2000 kDa, 24 wt.% in H2O; Lupasol SK) were kindly supplied by BASF SE (Ludwigshafen, Germany). Partially (80%) ethoxylated polyethyleneimine (PE-PEI; MW ~70 kDa, 35–40 wt.% in H2O) was purchased from Sigma Aldrich (St. Louis, MO, USA). Sodium hydroxide (for analysis; Merck, Darmstadt, Germany) and nitric acid (65%, for analysis; Merck) were used to adjust the working pH of the feed solution.

### 2.2.2. Ultrafiltration Setup and Analytical Methods

The ultrafiltration system consisted of a peristaltic pump (Ismatec, Zürich, Switzerland), an ultrafiltration module (Sartorius, Göttingen, Germany), a pH electrode (SI Analytics, Mainz, Germany), and a pH meter (Schott, Mainz, Germany) (Figure S1). The feed solution was magnetically stirred. Vivaflow® 50 filtration units (Sartorius) consisting of a flat membrane made of polyether sulfone (molecular weight cut-off (MWCO) = 5 or 10 kDa, A = 50 cm2) were used as ultrafiltration modules. During the ultrafiltration process, different material flows appear, while the feed solution is piped through the ultrafiltration membrane. Permeate is defined as the flow passing the membrane, whereas retentate is the flow retained by the membrane. Sampling of feed (1, Figure S1) and permeate (2, Figure S1) was performed 15 min after each pH adjustment. All permeate and feed samples were analyzed by ICP-OES. The polymer concentration in permeate and feed was determined by total organic carbon (TOC) measurement using a TOC analyzer (Shimadzu, Kyoto, Japan). Calibration of the latter was performed by determining the TOC of various concentrations of the respective polymer in ultrapure water. The resulting metal and polymer retentions were calculated using Equation (1):

$$R\_i \text{ [\%]} = (1 - (\text{c}\_{p,i}/\text{c}\_{f,i})) \times 100 \tag{1}$$

where *i* is the metal species or polymer, R*<sup>i</sup>* is retention of *i*, c*p*,*<sup>i</sup>* is concentration of *i* in the permeate, and c*f*,*<sup>i</sup>* is concentration of *i* in the feed.
