11institutetext: 1University College London, 2Adobe Research, 3University Of
Montreal

# Temporal Residual Jacobians for
Rig-free Motion Transfer

Sanjeev Muralikrishnan1 Niladri Dutt1 Siddhartha Chaudhuri2 Noam Aigerman3
Vladimir Kim2 Matthew Fisher2 Niloy Mitra1,2 Sanjeev Muralikrishnan 11
0000-0002-3556-5007 Niladri Dutt 11 0000-0002-7423-2221 Siddhartha Chaudhuri
22 0009-0009-8588-1436 Noam Aigerman 33 0000-0002-9116-4662  Vladimir Kim 22
0000-0002-3996-6588 Matthew Fisher 22 0000-0002-8908-3417 Niloy J. Mitra 11
0000-0002-2597-0914 22

###### Abstract

We introduce Temporal Residual Jacobians as a novel representation to enable
data-driven motion transfer. Our approach does not assume access to any
rigging or intermediate shape keyframes, produces geometrically and temporally
consistent motions, and can be used to transfer long motion sequences. Central
to our approach are two coupled neural networks that individually predict
local geometric and temporal changes that are subsequently integrated,
spatially and temporally, to produce the final animated meshes. The two
networks are jointly trained, complement each other in producing spatial and
temporal signals, and are supervised directly with 3D positional information.
During inference, in the absence of keyframes, our method essentially solves a
motion extrapolation problem. We test our setup on diverse meshes (synthetic
and scanned shapes) to demonstrate its superiority in generating realistic and
natural-looking animations on unseen body shapes against SoTA alternatives.
Supplemental video and code are available at
https://temporaljacobians.github.io/.

##  1 Introduction

A major challenge in character animation is to transfer the motion of a source
(skeletal) system to a diverse range of target characters in a realistic
manner. The traditional approach to achieve this involves using a rig, which
connects a skeleton to the character’s surface and manages the surface motion
through a variety of constraints and parameters. Target movement is then
conveyed from the skeleton to the surface by either simulating the physics of
the muscles and fat or by using tailored sets of skinning weights. These
weights can be manually created by skilled artists or derived from pre-
existing rigged models [37, 36]. Despite its simplicity, rigging can be time
consuming to set up and complicated to transfer to new target shapes; it may
also fail to accurately capture dynamics.

There has been growing interest in developing surface deformation techniques
that are more flexible and efficient than traditional rigging-based approaches
and can utilize available volumes of full-body motion capture data. One
possibility is to train data-driven methods to learn a low-dimensional
parameterization, such as a morphable humanoid template [20] or a neural space
deformation [38], to provide controllable handles for shape and pose-aware
manipulations. However, such methods do not capture continuity over time and
can overlook subtle motion dynamics essential for enhancing the realism of the
generated motion sequences. To add time-dependent effects, corrective vertex
deformations, similar to DMPLs [20] and SoftSMPL [31], have been introduced in
multistage workflows. Unfortunately, such methods do not account for
elementwise temporal inter-relations and have limited generalization to unseen
characters.

Figure 1:  Given a stick figure dance motion (top-right), Temporal Residual
Jacobians retarget the animation to unseen, unrigged meshes (top-left) across
time, producing realistic motion dynamics. Please refer to the supplemental
webpage for videos. Our method can be trained on limited data, does not
require rigged models or skinning weights during training or inference, and
does not assume paired sequences or registration to any canonical template
mesh. The method was trained on other bodyshapes: no target characters were
seen during training. All results in the paper and supplemental material were
obtained with automatic feature correspondences and without any postprocessing
or smoothing applied.

We aim to transfer a source motion, expressed as joint angles on a stick
figure, onto a target shape, specified as a (rig-free) mesh. We want to do so
without access to any rigging on the target shape, and we also want to avoid
any fixed template or morphable shapes. A desirable solution should address
several challenges: (i) handle rig-free meshes and/or scans with arbitrary
topology; (ii) produce plausible transfers to diverse shapes; (iii) achieve
continuity over space and time and thus avoid artifacts (e.g., broken meshes,
jittery motion); and (iv) work with long motion sequences without significant
drift. The first two problems are partially handled by rig-free pose transfer
(i.e., transferring a pose to a target mesh) methods [2, 19, 34]. While such
methods produce plausible single frames, they lead to jittery motion transfer
and motion-induced geometric artifacts like shearing.

We propose a novel approach that learns local spatio-temporal changes to
produce natural-looking motion transfers. Technically, we achieve this via a
new representation in the form of Temporal Residual Jacobians that temporally
links spatial predictions and is directly supervised using example motion
sequences. We jointly train two neural networks to individually predict local
spatial and temporal changes. They are coupled by spatial integration with a
differentiable Poisson solve, and temporal integration with a neural ODE. A
key technical insight is that instead of having the neural ODE predict per-
frame mesh deformations, it is more effective to predict initial deformations
independently of time via a base posing model (we use Neural Jacobian Fields
[2]), and then have the neural ODE predict residual deformations, linked over
time, as corrective factors that improve temporal coherence. Figure 1 shows
how a stick-figure control motion produces target motions for different
characters, without requiring registration to any standard template or
character rigs.

We evaluate the effectiveness of our method for motion generation to different
character bodies (e.g., humanoids, animals, Mixamo characters, scans) and
different motions (walk, run, jump, punch, dance). We compare our approach to
alternatives, when available. We provide qualitative and quantitative results
using the AMASS [21], COP3D [32], and 4DComplete [17] datasets.

In summary, our primary contributions are: (i) a novel method that enables
motion transfer via Temporal Residual Jacobians and can be trained directly
using positional data; (ii) local predictors that can be integrated in space
and time to create natural looking character animations; and (iii) a robust
pathway to transfer realistic character motion without the need for explicit
rigging or learning a parameterization using any canonical template shape.

##  2 Related Work

#### Parametric Shape Deformation.

These methods express 2D or 3D shapes as a known function of a set of common
parameters, and model deformations as variations of these parameters. Such
methods include cages, explicit [14] or neural [38], blendshapes [15], skinned
skeletons [13], Laplacian eigenfunctions [30], and several other variations.
Linking the parameters to the shape’s surface often requires manual annotation
of weights (commonly known as weight painting) in 3D authoring tools.
Alternately, given sufficient data (i.e., meshes, rigs, skinning weights),
end-to-end training can produce realistic neural rigs, as demonstrated by
Pinocchio [4], RigNet framework [36], skinning-based human motion retargeting
[22], and skeletal articulations with neural blend shapes [16]. Unsupervised
shape and pose disentanglement [42] proposes to learn a disentangled latent
representation for shape and pose, which can be further used to transfer
motion using shape codes. This requires meshes to be registered and have the
same connectivity. To animate these parametrized shapes through time, the
parameters are varied over time and the dynamic weights animate the mesh.
These methods require access to body templates and/or rigs and can produce
results that are jittery due to loose coupling of the individual frame
predictions.

#### Dynamic Motion.

It is possible to model temporal surface effects by simulating the underlying
soft tissues using finite element methods (FEM) [6, 9]. This direct simulation
is typically slow and requires artists to design the underlying bone and
muscle structure [1]. Approaches have been developed to overcome the stiffness
problem in FEM to accelerate simulating these systems [23] or to solve the
problem in a lower-dimensional subspace [25]. For the specific case of rigged
human characters, Santesteban et al. [31] add soft-tissue deformation as an
additive per-vertex bump map on top of a primary motion model; AMASS [21]
imparts secondary motion using the blending coefficients of the DMPL shape
space [20]; and Dyna [26] learns a data-driven model of soft-tissue
deformations using a linear PCA subspace. These efforts, however, assume
access to primary motion via a skeleton rig and are restricted to humans,
registered to a canonical template. Complementary dynamics [40, 5] models
physically-based secondary motion in the subspace complementary to that
spanned by an animation rig. This approach adds automatic secondary motion to
arbitrary animated objects, but requires the target shape to be rigged, is not
designed for deformation transfer (the base animation is part of the input),
and is tied to a specific hand-coded secondary physics model. DeepEmulator
[41] achieves a similar effect using a local-patch-based neural network to
learn the secondary behavior, but again requires the primary motion as input
and does not support deformation transfer.

#### Discrete Time Motion Models.

Given their ability to model time, deep recurrent neural networks have also
been used to model shape sequences. Fragkiadaki et al. [10] use LSTMs to
predict short human joint motions given initial frames. Harvey et al. [11]
leverage LSTMs for in-betweening to predict intermediate joint motions. He et
al. [12] learn a motion field for joints through time. In all these methods,
the mesh itself is deformed via rigging, and since joint motion does not
encode bodyshape, they do not sufficiently represent secondary motion. Also,
being discrete time representations, these approaches must train on large
datasets of joint motion. Qiao et al. [27] instead use mesh convolutions with
LSTMs to deform vertices through time. In our work, we use neural ODEs as they
can model time continuously instead of discretizing time and modeling the
sequence using LSTMs. Further, vertex-based deformation models are susceptible
to artifacts like normal-inversion as we demonstrate in our evaluation
section.

##  3 Approach

Given an unrigged, triangulated mesh of a 3D character, we aim to animate it
by motion transfer from an available motion described by relative joint angles
(stick figure motion) at each time step. The relative joint angles are
represented as Euler angles and are defined at each joint with respect to its
hierarchy in SMPL’s kinematic tree (cf. [12]). We obtain these joint-based
motion representations from the AMASS dataset [21]. Since we aim to animate
the mesh itself from the joints’ motion, we seek to learn a mapping from the
joint representation to the positions of the given mesh’s vertices, and to do
so at each time step while ensuring we generate a smooth and artifact-free
mesh animation. We supervise our setup with ground-truth meshes from the AMASS
dataset [21].

We parameterize such a character $X$ $\in\mathbb{R}^{N\times 3}$ as assigning
positions to each of its $N$ vertices of the underlying mesh. Thus to impart a
motion to a mesh, we perform this assignment at every time step. Given a shape
$X_{0}$ as a triangulated mesh, in its initial pose configuration, along with
per-frame pose configuration $M_{t}$ that describes the target pose at time
$t$, we aim to predict the shape $X_{t}$ at each time $t$ generating the full
motion sequence; essentially learning a mapping from relative joint
orientations to mesh deformations (typically, from 30-50 joints to 50-100k
vertices). We aim for a data-driven method that generalizes to new characters
and does not rely on shape rigging or intermediate shape keyframes. Another
desirable property is to do so with limited data (e.g., working with 5-10
motion examples), as obtaining full-body 3D motion data is non-trivial.

Our key observation is that we can robustly train neural networks to predict
changes local in both space and time, which can then be integrated across
space, using a differentiable Poisson solve; and across time, using Euler
equations to handle an ODE numerically. Integrating across space helps
maintain plausibility of the entire shape, while integrating across time helps
model a realistic, time-coherent motion. Further, local predictions help
higher generalization capability to unseen body shape; while the spatio-
temporal integration ensures smoothness of the sequence, without jitters or
undesirable shearing artifacts. Our end-to-end differentiable formulation
allows training the networks directly using example training sequences,
without requiring factorized spatial and temporal motion signals. For spatial
handling, we extend the representation from local deformation encoding [33]
and affine mapping framework Neural Jacobian Fields (NJF) [2], which learns an
affine transformation field that is sampled at each mesh face and integrated
into vertex positions via a Poisson solve. For temporal coherence and
consistency in these predicted Jacobians, we couple temporal signals across
the character motions using a neural ODE framework [7] via a novel Temporal
Residual Jacobian representation. Predicting residual deformations that
correct the predictions of a base model for time-coherence turns out to be
more effective than trying to make the base model itself time-coherent.

###  3.1 Overview

#### Training:

Since our goal is to map temporally varying joint angles to full mesh
animation, during training we assume this mapping is available i.e., we have
one-to-one mapping across time between joint orientations and the
corresponding meshes. Thus, we use the motion sequences in AMASS [21] as our
training dataset for humanoid shapes. Thus, during training, our algorithm
takes in the joint angles at each time step, the mesh itself at time $t=0$ in
its starting pose, and the full mesh sequence is used for supervising our
neural networks.

#### Inference:

At inference time, our algorithm simply takes in an unrigged character
provided as a triangulated mesh. Our trained method is then used to animate
this mesh from joint-sequences of motions sampled from AMASS. These meshes can
be obtained in-the-wild or from character datasets like Mixamo. For shapes in
the AMASS dataset since we have the ground-truth motions, we evaluate our
framework quantiatively. We only show qualitative motion generation for other
shapes in the work, as there is no ground-truth solution.

#### Motion and Shape parameters:

The motion sequences in AMASS were obtained from live captures of subjects
performing different motions; these were then parametrized in SMPL’s [20]
shape and pose space, thus providing a mapping from joint orientations to 3D
mesh pose - i.e., vertices of the mesh oriented to match the pose represented
by the provided joint orientations. We use this mapping to supervise our
framework. Specifically, this data framework allows us to sample motions
across both shapes and poses, by varying the respective low-dimensional
parameters $\beta$ and $\alpha$. In our work, the $\alpha$ translate to joint
angles at each time step and we use the source live-capture subject’s $\beta$
as is, and pass it as a conditioning variable to our method.

###  3.2 Preliminaries

Our framework uses two components – one to learn spatial mesh deformations
(re-posing) and the other to learn temporal signals to generate a temporally
coherent mesh motion. We describe these below.

#### Neural Jacobian fields,

as introduced in [2], encode local deformations by defining a field via map
$\phi$, defined over space, that can be sampled at the $N$ vertices of the
surface mesh. Specifically, given $\phi$, we compute Jacobians in the basis of
the triangles of the surface mesh as,

$J_{i}=\phi\nabla_{i}^{T}$ (1)

where $\nabla_{i}$ is defined as the gradient of triangle $t_{i}$ in its basis
$B_{i}$ defined at the triangle’s centroid. Thus, given a learned map, we
obtain each triangle’s (estimated) affine transformations $J_{i}$. Recovering
the vertex positions from this per-triangle assignment of affine
transformations is then done via a least-squares formulation that reduces to
solving a Poisson system given by (cf. [39]),

$\phi^{*}=L^{-1}\rho\nabla^{T}J$ (2)

where $\rho$ is the mesh’s mass matrix, $L=\nabla^{T}\rho\nabla$ is the
cotangent Laplacian, and $J$ is a stack of (estimated) Jacobians $J_{i}$. This
solution gives a unique mesh up to a translation, which we fix using $X_{0}$,
the mesh at $t=0$. Mesh positions directly supervise the neural map $\phi$ via
a differentiable Poisson solver [24].

Thus, NJF allows us to learn affine deformations of a given mesh. In our work,
the map $\phi$ is a trainable neural network that predicts local deformations
of the mesh’s triangles. Given conditioning parameters, the trained map can
then be used to predict the Jacobians of each triangle in the mesh at each
time step, generating a time-coherent animated 3D mesh.

Figure 2:  Method overview.  Starting from input stick figure motion
($\\{M_{i}\\}$) and a target body shape ($X_{0}$), Residual Temporal Jacobians
makes local predictions, using primary $f_{P}$ and residual $f_{R}$ MLPs, to
predict spatial and temporal changes to per-triangle Jacobians. These are then
integrated in space, via a Poisson solve, and in time, via numerical Euler
stepping, to predict motion dynamics at time frame $t$. These two learnable
modules are trained simultaneously with only direct object-level supervision
using a combination of positional and Jacobian losses. We use the ground-truth
meshes from AMASS to supervise the predictions. The time $t$ input is
positionally encoded.

#### Neural ODEs

Chen et al. [7] represent differential equations with neural networks to model
the dynamics of time-varying systems (motion sequences in our setting).
Specifically, given a function $f(t;\theta):=\frac{\partial J}{\partial t}$,
where $f$ is a neural network with parameters $\theta$ and $t$ being time, the
variable $J$ can be integrated out at $t$ as

$J(t)=\int_{0}^{t}\frac{\partial J}{\partial
t}dt=\int_{0}^{t}f(t;\theta)dt+J_{0}.$ (3)

The neural network can be directly supervised with $J$ values as the network
output utilizes a black box differential equation solver without explicitly
discretizing the dynamic system. This leads to better estimation, benefits
from constant memory updates (as opposed to explicit hidden states in RNNs,
LSTMs, etc.), and trades numerical precision for speed.

Our work seeks to learn how the triangle Jacobians move through time. Thus,
our neural ODE operates in the Jacobian space, with Jacobians being the
variables to integrate across time.

###  3.3 Motion Transfer with Space-time Integration

Without keyframes, we have to solve a temporal extrapolation problem. Our goal
is to produce shape-preserving realistic motion sequences, as well as to learn
such a motion model from a very sparse dataset. Not surprisingly, if each
frame were independently predicted, the resulting motion would not be
consistent across time, resulting in a jittery and artifact-ridden sequence
(see Section 4). We, therefore, propose a novel framework wherein independent
frame predictions are improved by providing temporal signals from a neural
ODE. The two networks work in lockstep, each boosting the other’s predictions,
and are trained together.

#### Independent posing:

We estimate each frame’s Jacobians independent of temporal information. Given
a shape $X_{0}$, pose configuration (relative joint angles with respect to
SMPL’s kinematic tree) $M_{t}$ and features describing each mesh face, we
predict $J^{P}_{t}$ as the primary Jacobians at time $t$. Specifically,

$\displaystyle\Delta J^{P}_{t}$ $\displaystyle=$ $\displaystyle
f_{P}(J^{P}_{0},M_{t},C;\theta_{P})$ (4) $\displaystyle J^{P}_{t}$
$\displaystyle=$ $\displaystyle J^{P}_{0}+\Delta J^{P}_{t}$ (5)

where $J^{P}_{0}$ is the first frame Jacobian computed from $X_{0}$, $M_{t}$
is the pose configuration given as relative joint angles, and $C$ are per-face
features defined at the centroid of the mesh faces, of the mesh at $t=0$. In
our tests, as features $C$, we jointly learned PointNet features on the
centroids and normals of $X_{0}$ and augmented them with pre-computed Wave
Kernel Signatures [3]. Our feature network (a shallow PointNet) learns
geometric features that enable mapping to an unseen shape via correspondence
in this learned feature space. We note that conditioning the posing network on
$J^{P}_{0}$ and adding the predicted delta via a residual connection
significantly improves pose prediction and generalization to unseen pose
configurations. We additionally note that the Jacobians are computed in the
local basis defined at each face’s centroid. Thus, $J^{P}_{0}$ is the Jacobian
of the identity deformation in the local basis (a rotation). Henceforth, we do
not update the basis in the sequence and express all Jacobians on this chosen
basis.

In practice, we analytically compute the triangle Jacobians of only the first
frame $X_{0}$ in the sequence. We then predict the Jacobians at each time
instance $t$ in the coordinate frame of $X_{0}$ and solve Equation 2 to obtain
the shape $X_{t}$ at time $t$. Importantly, we augment NJF with temporal
learning signals, by linking these independent per-frame predictions via a
neural ODE, as described next.

#### Stitching across time:

Our key observation is that learning local changes in time generalizes better
to unseen shapes. Central to our method is a neural ODE that provides temporal
training signals to the primary NJF and integrates across time to learn a
smooth, arbitrary-length motion sequence. We found independent per-frame
predictions are prone to artifacts and do not generalize well (see Section 4).
We also observe that a neural ODE, in isolation, cannot predict the entire
sequence due to the drift problem inherent to estimating functions using
numerical ODE methods. Specifically, given an initial state and per-frame
control parameters (in our case, joint angles), predicting the mesh sequence
is an extrapolation problem. As such, ODEs are prone to drifting away from the
underlying function. This problem is exacerbated as the length of the motion
increases – the longer the motion, the larger the accumulated drift.

As a solution, we propose a novel formulation to address both the incoherence
of per-frame predictions and the drift problem. Specifically, we direct the
neural ODE to learn only Residual Jacobians at each time step conditioned on
the predictions from Eq 5 and on a window of past Jacobians. The Residual
Jacobians are corrective factors which are directly added to the per-frame
Jacobians. We predict Residual Jacobians local in time as outputs of a Neural
ODE to ensure temporal coherence. This allows us to handle much longer
motions, spanning 1-3k frames, without noticeable drift. We describe our
method below.

#### ODE formulation:

To handle arbitrary length sequences, in the interest of memory and training
speed, we train and infer a given sequence in windows of consecutive frames. A
given sequence is broken into fixed window sizes. We initialize $J_{0}$ in Eq
3 as the first frame’s Residual Jacobian. Since the first frame is stationary
and given as input, we set the first frame’s Residual Jacobian as,

$J^{R}_{0}=\textbf{0}\in\mathbb{R}^{3\times 3}.$ (6)

We then task the neural ODE to learn Residual Jacobian $J^{R}_{t}$ at each
$t$, by extrapolating from $J^{R}_{0}$ using Euler’s integration. We then
extract the final Jacobians in terms of the base Jacobians corrected by the
Residual Jacobians as

$J_{t}=J^{P}_{t}+J^{R}_{t}.$ (7)

We predict the residuals $J^{R}_{t}$ by integrating over time the bodyshape-
specific local changes predicted by $f_{R}$ which is defined as,

$\frac{\partial J^{R}_{t}}{\partial
t}=f_{R}(J^{P}_{0},E^{P}_{W},E^{R}_{W-1},\beta,t;\theta_{R})$ (8)

where $\beta$ is the shape signature that defines the given body shape, and
$J^{P}_{0}$, as defined previously, are the Jacobians of the first frame;
$E^{P}_{W}$and $E^{R}_{W-1}$are attention encodings of current-window pose
predictions and previous-window residual predictions. We integrate the local
changes (see Eq 8) over time using Euler’s method to obtain $J^{R}_{t}$ at
each $t$ as,

$J^{R}_{t}=\int_{0}^{t}\frac{\partial J^{R}_{t}}{\partial
t}dt+J^{R}_{0}=\int_{0}^{t}f_{R}(J^{P}_{0},E^{P}_{W},E^{R}_{W-1},\beta,t;\theta_{R})dt.$
(9)

Our jointly trained attention encoders are defined as,

$\displaystyle E^{P}_{W}$ $\displaystyle=$ $\displaystyle
A^{P}(J^{P}_{W},T_{W})$ (10) $\displaystyle E^{R}_{W-1}$ $\displaystyle=$
$\displaystyle A^{R}(J^{R}_{W-1},T_{W-1})$ (11)

where $A^{P}$ and $A^{R}$ are multi-head attention networks, $J^{P}_{W}$ and
$J^{R}_{W-1}$ are a block of sequential Jacobians in the current window $W$
and past window $W-1$, respectively; $T_{W}$ and $T_{W-1}$ are correspondingly
blocks of time instances in these windows and are positionally encoded using
time. In all our experiments we use a window size of 32 frames.

Note that we use the attention networks to encode a window of Jacobians to a
single encoding. Since the encoding sizes are thus constant, we can handle
arbitrary window/sequence lengths without overflowing memory. These encoders
distill the current posed Jacobian predictions from Eq 5 and previously
predicted Residual Jacobians obtained from Eq 9.

We pass the output of the attention networks as conditioning to Eq 8 to
integrate and obtain the residuals in Eq 9. The predicted residuals are then
added to the posed Jacobians in Eq 7. Finally, we spatially integrate the
predicted Jacobians $J_{t}$ using a differentiable Poisson solve [24], in the
coordinate frame of the first frame, to obtain the predicted shape $X_{t}$ at
$t$.

#### Loss function:

Our pipeline is trained end-to-end using only a shape loss over vertices of
$X_{t}$ and a Jacobian loss. Thus, our final objective function is simply

${}\mathcal{L}_{\text{vertex}}=\|X_{t}-X_{t}^{GT}\|^{2}\quad\text{and}\quad\mathcal{L}_{\text{Jacobian}}=\|J_{t}-J_{t}^{GT}\|^{2},$
(12)

$\mathcal{L}=\mathcal{L}_{\text{vertex}}+\alpha\mathcal{L}_{\text{Jacobian}}.$
(13)

We use $\alpha=0.05$ in our tests.

Figure 3:  Generalization across bodyshapes. We show results of different
motion transfers on meshes found in-the-wild (blue), FAUST scans (pink) and
Mixamo characters (green). We observe a smooth motion consistent with the
target geometry in each case. Please see supplemental materials. Figure 4:
Generalization across shapes with very sparse training sets. Here, we show
motion transfer from two animal sequences (in yellow) sampled from the
DeformingThings4D dataset [18], to animal meshes found in the wild (in blue).
Our method was trained on only two sequences from this dataset and yet
generates plausible motion transfer to unseen shapes. Rigs were not available
to our algorithm at training and/or test time. (Note: blue sequences have been
slightly globally rotated for visibility.)

##  4 Evaluation

#### Motion datasets.

We train and evaluate our method on motion sequences from three different
datasets. First, the AMASS dataset [21] for humanoid motions, which is based
on the SMPL body model [20] that allows us to vary body-shapes by changing its
shape parameter $\beta$, for a given sequence. Thus, we train our model for a
given motion category (like running) to predict the sequences on varying body-
shapes (indicated as the $\beta$-predictor in Figure 2). During training, each
sequence in the category is trained with only one body-shape. During
inference, given an unknown test shape $X_{0}$ and a motion $\\{M_{t}\\}$, we
synthesize the full body motion for the target shape. Second, we also tested
on a synthetic DeformingThings4D dataset [18], which provides animal 4D meshes
as deforming sequences. Finally, we also present results on motions extracted
from video recordings of animals in the COP3D dataset [32] where our inputs
are meshes fitted to the video recordings to train Temporal Residual
Jacobians.

#### Target shapes.

We evaluate our method on various forms of target shapes, namely sampled SMPL
models, scans from the FAUST dataset, characters from the Mixamo library,
various models from online 3D repositories (e.g., wolf, triceratop, monster,
hole-man, etc.). Non-manifold inputs were converted to manifold meshes before
running our algorithm. All correspondences were as inferred by the signature
module (i.e., combination of PointNet features on centroids and normals of
faces along with WaveKernel Signatures, previously defined as $C$).

#### Implementation details.

All the networks we use are shallow MLPs to aid training speed. For our
independent posing network, we use a 4-layer MLP with ReLU activation, with
the final layer being linear. For the residual Jacobian prediction, we use a
3-layer MLP similarly with intermediate ReLU activations and a final linear
layer. Both our attention networks follow the same architecture – we use two
attention heads, with a 32-neuron wide feed forward layer and 32-dimensional
features for the keys and values. Our PointNet network from which we obtain
per-face features similarly has 3 ReLU layers followed by a linear layer at
the end. Our method and the baselines are trained until
$\mathcal{L}_{\text{vertex}}$ converges to less than 3e-4 or upto 300
iterations. On a 12 GB TitanX our typical training time is 6-8 hours, varying
by the lengths of the sequences.
For the walking and dancing motions, we set the root orientation in AMASS to
zero, so all subjects are front facing, making it easier to train (similar to
[12]), as our Jacobian-based formulation cannot, on its own, infer global
rotation and/or translation. We do not, however, predict global
transformations; we obtain the global transforms from the source AMASS
sequences and apply them to our final outputs at inference after appropriate
scaling according to the target’s height.

#### Qualitative results.

We show various examples of deformation transfer by our method in Figures 1,
4, and 5. Please refer to the videos in the supplemental webpage. Our method
generalizes to new shapes of varied body types, including non-humans
completely unseen during training, while preserving the source motion. We show
examples on monsters, a 4-armed character, etc. Additionally, as different
humans perform the same motion in different manners, we capture those varying
dynamics in the deformation of the new shape as well. We can also learn and
apply motions from animals, both from synthetically generated mesh sequences
as well as motions extracted from monocular video recordings.

Figure 5:  Motion transfer from COP3D dataset. We train on only four sequences
of dogs obtained from the COP3D dataset [32], which are monocular video
recordings of animal motion, and transfer the observed regressed motion (in
yellow) to creature meshes found in the wild (in blue).

#### Comparisons.

Our method Temporal Residual Jacobians, due to the space-time coupled
formulation, is designed to produce natural-looking motion retargeting.
Although we are unaware of any other method that performs the specific motion
extrapolation problem (i.e., without access to keyframes), we compare with
possible baseline design variations. First, VertexODE, where an MLP predicts
the rate of change of vertex positions (velocity) at time $t$, followed by a
Neural ODE [7] that takes numerical steps by Euler’s method to integrate to
the vertex positions at $t$. This baseline directly displaces the vertices
(i.e., without triangle Jacobians). As seen in Figure 6, this method produces
significant artifacts and leads to degenerate shapes. It also loses the
intended motion after some time, as seen in the videos. Second, NJF($M_{t}$),
where we extend the original NJF framework to be additionally conditioned on
per-frame relative joint orientations $M_{t}$, to predict the Jacobians at
each time step. Framewise prediction leads this approach to suffer from
jitters (please refer to supplemental videos) and a tendency to overfit to
training data. Although the individual frames are mostly plausible, they
occasionally suffer from frame-level artifacts as shown in Figure 6. The
primary limitation, however, is jittery output as seen in the video results
since the frames are independently generated. Note that a method without
spatial or temporal derivatives wherein a simple MLP is trained to predict
vertex deformations given the initial geometry and time $t$ performs poorly,
with numerous artefacts due to vertex-level predictions.

Figure 6:  Observed artifacts in baselines. For each motion, we show results
from an intermediate frame of the motion transfer for our baselines. VertexODE
(left, yellow) completely distorts the shape, while not following the target
motion. NJF (right, brown) suffers from temporal discontinuity resulting in
motion-driven geometric artifacts – extended or shrunken parts as it tends to
a linear path in later time steps – and jitter.

#### Quantitative comparison:

We evaluate our method on several types of sequences and compare ours against
other baselines in Table 1 by computing the residual error with respect to the
ground-truth sequences. We compare five different motion categories from the
AMASS dataset - running, jumping jacks, walking, hopping, and punching. We
achieve the best on all metrics.

#### Human motion to non-humanoid characters.

We show results on human motion applied to non-humanoid characters (e.g.,
4-armed monster, alien-reptile) in supplemental videos. Please note the
quality of automatic transfer without manual landmark specification.

Table 1:  Quantitative evaluation. Average vertex-to-vertex error in cm, L2 error of predicted Jacobians and angular error of normals in degrees, measured against ground truth sequences, for different motion categories and averaged over multiple sequences within the same target motion category. Here we compare against neural ODE [7] and an extended version of NJF [2]. Lower values indicate better generalization. Method | Jump | Run | Punch | Walk | Dance
---|---|---|---|---|---
L2-V | L2-J | L2-N | L2-V | L2-J | L2-N | L2-V | L2-J | L2-N | L2-V | L2-J | L2-N | L2-V | L2-J | L2-N
VertexODE | 22.59 | 1.22 | 48.21 | 14.23 | 0.83 | 42.11 | 17.66 | 0.65 | 40.04 | 23.92 | 1.01 | 46.12 | 26.15 | 1.26 | 50.17
NJF($M_{t}$) | 5.52 | 0.41 |  9.66 | 3.61 | 0.32 | 7.38 | 4.95 | 0.38 | 7.42 | 6.85 | 0.34 | 8.12 | 4.86 | 0.44 | 11.24
TRJ (Ours) | 2.64 | 0.28 | 7.31 | 1.73 | 0.24 | 5.63 | 2.86 | 0.26 | 6.65 | 1.48 | 0.22 | 6.33 | 3.96 | 0.37 | 9.88

#### Handling long sequences.

One advantage of our space-time coupled formulation is the ability to handle
long motion sequences. We show examples of motion transfer from a few hundred
to a couple of thousand frames (dance and walk) in the supplemental.

#### Design variation.

A seemingly possible variation of ours is to choose a residual Jacobian
representation, but expressed in terms of the previous predicted frame. While
this seems attractive, we found it very slow to train as we have to
backpropagate through time, and convergence is very slow (when using moderate
computational resources). Hence, we found this approach to be infeasible with
the current memory requirements for attention (and transformer) modules.

#### Alternatives to Euler Solve.

We experimented with higher-order ODE solvers, namely Runge-Kutta and Runge-
Kutta-Fehlberg. However, we noticed no significant improvements, even with the
noticeably increased training time. We use the simpler Euler’s method for
temporal integration for our tests.

##  5 Conclusion

We have presented Temporal Residual Jacobians, a spatially-coupled, neural
ODE-based, motion transfer framework conditioned on body type and target
motion to produce local Jacobians that are subsequently integrated across
space and time to deform the target geometry. The resultant motions are
robust, realistic, and generalize to different body types. We extensively
tested our method on both synthetic and real data captures, and enabled
generic motion transfers to an extent which is not possible using existing
methods.

Our method has limitations. (i) We do not impose physics constraints,
therefore, our animation can have self-intersections (see webpage videos). An
interesting direction is to incorporate constraints for collision detection,
e.g., via subspace-based contact handling [29]. We want to incorporate such an
approach directly into the method, possibly via an attention mechanism, as
motion dynamics are affected by earlier collisions. (ii) Although our Temporal
Residual Jacobians, along with windowed attention modules, keep drifts low,
error still accumulates over long motion sequences. A possible solution is to
couple our method with a keyframe-based workflow [35]. (iii) Finally, our
current formulation implicitly establishes correspondences and uses them to
infer temporal Jacobians for surface triangles. This step may incur errors as
shown in project webpage (e.g., flapping ear of the mouse). These spurious
correspondences can be overridden by artists, possibly directly by
“paintbrushing” correspondences or materials (e.g., indicating that the ear of
the bunny model should be floppy) or using semantic features learned from
untextured meshes [28, 8].

Acknowledgement This project has received funding from the UCL AI Center,
gifts from Adobe Research, and EU Marie Skłodowska-Curie grant agreement No
956585.

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# Microfluidic osmotic compression of a charge-stabilized colloidal
dispersion:
Equation of state and collective diffusion coefficient

Camille Keita1 Yannick Hallez2 Jean-Baptiste Salmon1 1CNRS, Solvay, LOF, UMR
5258, Univ. Bordeaux, F-33600 Pessac, France  2Laboratoire de Génie Chimique,
Université de Toulouse, CNRS, INPT, UPS, Toulouse, France

###### Abstract

We show, using a model coupling mass transport and liquid theory calculations
for a charge-stabilized colloidal dispersion, that diffusion significantly
limits measurement times of its Equation Of State (EOS), osmotic pressure vs
composition, using the osmotic compression technique. Following this result,
we present a microfluidic chip allowing one to measure the entire EOS of a
charged dispersion at the nanoliter scale in a few hours. We also show that
time-resolved analyses of relaxation to equilibrium in this microfluidic
experiment lead to direct estimates of the collective diffusion coefficient of
the dispersion in Donnan equilibrium with a salt reservoir.

The dynamics of many out-of-equilibrium processes involving colloidal
dispersions, such as drying, ultrafiltration, or sedimentation rely primarily
on two fundamental inputs: the Equation of State (EOS) osmotic pressure $\Pi$
vs colloid concentration $\varphi$, and the collective diffusion coefficient
$D(\varphi)$ relating the concentration gradient to the colloid flux in Fick’s
law [1, 2, 3, 4, 5, 6, 7]. In ultrafiltration for example, the colloid
accumulation at the membrane, the so-called concentration polarization, limits
the process by the local increase in osmotic pressure, but is limited by
collective diffusion that opposes the formation of concentration gradients
[8]. These quantities are also central to many fundamental questions, such as
the validity of a generalized Stokes-Einstein relation for colloidal
dispersions [9, 10, 11]. Beyond dispersions, these quantities, and more
particularly the EOS $\Pi$ vs $\varphi$, are also fundamental to probe and
understand interactions of soft matter systems at colloidal scales (see e.g.,
Refs. [12, 13, 14]).

For charge-stabilized dispersions, $\Pi(\varphi)$ and $D(\varphi)$ depend on
colloidal interactions [15] and thus on the ion concentration, and they are
difficult to measure. The EOS can be estimated by various techniques such as
analytical ultracentrifugation, vapor pressure or membrane osmometry [16], but
osmotic compression remains the primary technique because it allows one to
measure the EOS at fixed and imposed chemical potentials of salts in order to
control colloidal interactions. In this experiment [see Fig. 1(a)], a dialysis
bag (typical volume $\sim 10~{}$mL) filled with a dispersion is immersed in a
bath containing salts and polymers that impose an osmotic pressure, $\Pi_{i}$.
The bag is not permeable to polymers and colloids but allows the exchange of
solvent and ions to balance their chemical potentials, the so-called Donnan
equilibrium. The colloid concentration $\varphi^{\star}$ in the bag is then
measured (often by dry extract) leading to one point of the EOS, $\Pi_{i}$ vs
$\varphi^{\star}$, for the conditions imposed by the reservoir. For most
dispersions, the long equilibration times ($\geq$ days) prevent rapid
measurements of the EOS by osmotic compression and therefore its use in
particular in research and development to quickly screen formulations.
Moreover, this technique uses relatively large sample volumes, a few
milliliters per measurement point, and is therefore unsuitable for the
measurement of EOS of samples that are either expensive or available in too
small quantities.

Figure 1: (a) Osmotic compression for a charged dispersion. (b) and (c)
Concentration profiles evenly spaced from $\tilde{t}=0$ to $0.9$ for
$\text{Pe}=1$ and from $\tilde{t}=0$ to $0.014$ for $\text{Pe}=10^{3}$. (d)
Corresponding $R/R_{0}$ vs $\tilde{t}$; $\tilde{\tau}_{\text{eq}}$ values are
shown with bullets. (e) ($\bullet$) $\tau_{\text{eq}}$ vs $R_{0}$ estimated
from the numerical model. The dashed lines are the scaling laws with $R_{0}$
as explained in the text.

Measurements of $D(\varphi)$ are even more challenging for charged
dispersions. $D(\varphi)$ is often estimated from dynamic scattering
experiments whose clear interpretation is limited to monodisperse colloids
diluted enough to avoid multiple scattering [17, 18, 19, 20, 21, 22]. For less
monodisperse and more concentrated colloids, recent works reported estimates
of $D(\varphi)$ using direct measurements of both concentration gradient and
colloid flux in microfluidic experiments [23, 24]. Such experiments, however,
do not allow one to control the ionic concentration as for the Donnan
equilibrium, so that these measurements remain tricky to extrapolate to other
processes. In this work, we first use a model coupling mass transport and
liquid theory to highlight the limiting factor that makes osmotic compression
so slow for charged dispersions. Following this result, we then develop a
microfluidic chip to measure simultaneously the EOS $\Pi(\varphi)$ of a
charge-stabilized dispersion in a few hours and its collective diffusion
coefficient $D(\varphi)$, both in Donnan equilibrium.

Consider first a cylindrical dialysis bag of radius $R(t)$ and fixed length
$L\gg R$, filled by an aqueous dispersion of spherical colloids of radius $a$
carrying $Z$ unit charges [see Fig. 1(a)]. The concentration $c_{\text{res}}$
of monovalent salt in the bath is assumed to be fixed. The shrinkage rate of
the bag is given in this axisymmetrical model by [25]:

$\dot{R}(t)=\mathcal{L}_{p}\\{\Pi[\varphi(r=R,t)]-\Pi_{i}\\},$ (1)

where $\mathcal{L}_{p}$ is the permeability of the membrane. As $R$ decreases
over time, both colloid concentration and osmotic pressure increase and
equilibrium is reached for a homogeneous concentration $\varphi^{\star}$,
defined by $\Pi(\varphi^{\star})=\Pi_{i}$. Computing the time
$\tau_{\text{eq}}$ to reach equilibrium requires solving mass transport within
the bag. For this, we first assume that ions are always at equilibrium because
of their high diffusivity. For local thermodynamic equilibrium conditions
(ignoring, e.g., phase nucleation kinetics), assuming isothermal conditions at
room temperature $T$ and neglecting advection, the colloid volume fraction
follows:

$\frac{\partial\varphi}{\partial t}=\frac{1}{r}\frac{\partial}{\partial
r}\left(rD(\varphi)\frac{\partial\varphi}{\partial r}\right).$ (2)

For spherical and monodisperse colloids, $D(\varphi)$ is given by [26]:

$D(\varphi)=D_{0}K(\varphi)\chi_{\text{osm}}^{-1},$ (3)

where $D_{0}=(k_{B}T)/(6\pi\eta a)$ is the Stokes-Einstein coefficient ($\eta$
is the solvent viscosity and $k_{B}$ is the Boltzmann constant), $K(\varphi)$
is the long-time sedimentation hindrance function, and $\chi_{\text{osm}}$ is
the osmotic compressibility coefficient defined by:

$\chi_{\text{osm}}^{-1}=\left(\frac{\partial\beta\Pi}{\partial
n}\right)_{T,\,c_{\text{res}}}=\frac{1}{S(q\rightarrow 0)},$ (4)

where $n=\varphi/(4/3\pi a^{3})$, $\beta=1/(k_{B}T)$, and $S(q)$ is the
colloid structure factor. Note that both $\chi_{\text{osm}}$ and $K(\varphi)$
are evaluated at fixed reservoir conditions $c_{\text{res}}$. The boundary
condition:

$\varphi(R,t)\dot{R}+D[\varphi(R,t)]\frac{\partial\varphi}{\partial
r}(R,t)=0,$ (5)

ensures the rejection of the colloids by the membrane. As shown in the
Supplemental Material (SM) [27], this model can be made dimensionless defining
$\tilde{r}=r/R_{0}$ and $\tilde{t}=(D_{0}/R_{0}^{2})t$, with $R_{0}=R(t=0)$,
and depends on the Péclet number $\text{Pe}=R_{0}\mathcal{L}_{p}\Pi_{i}/D_{0}$
comparing the rate of bag shrinkage to colloid diffusion.

To solve this model, $\Pi(\varphi)$ and $D(\varphi)$ are evaluated as follows.
First, an effective interaction potential between colloids is designed as
follows:

$\beta
u=l_{B}Z_{\text{eff}}^{2}\frac{e^{-\kappa_{\text{eff}}r}}{r}\text{~{}for~{}}r\geq
2a,~{}\beta u=\infty\text{~{}for~{}}r<2a,$ (6)

where $l_{B}$ is the Bjerrum length, and $Z_{\text{eff}}$ and
$\kappa_{\text{eff}}^{-1}$ are respectively an effective charge and an
effective screening length obtained by the Extrapolated Point Charge (EPC)
renormalization method [28]. The Ornstein-Zernike (OZ) equation is then solved
for the structure factor $S(q)$ with either the Rescaled Mean-Spherical
Approximation or the HyperNetted-Chain closure, which yield indistinguishable
results for this system. $\Pi(\varphi)$ follows from the virial theorem [28].
$D(\varphi)$ is estimated using Eq. (3), assuming the long-time $K(\varphi)$
can be approximated by its short-time counterpart $K_{s}(\varphi)$ evaluated
using the renormalized density fluctuation expansion of Beenakker and Mazur
($\delta\gamma$ method) [29, 30, 31, 32, 33, 26, 34, 35, 36] taking as sole
input the structure factor $S(q)$. Any difference between $K_{s}(\varphi)$ and
$K(\varphi)$ would be due to non-pairwise hydrodynamic interactions,
negligible at small $\varphi$ [18]. As a relevant case, we consider $a=10$ nm,
$Z=450$, $T=20^{\circ}$C, $\eta=1$ mPas, and $c_{\text{res}}=10$ mM. The above
calculations give $D(\varphi)/D_{0}=3$–16 with $D_{0}\simeq 2.15\times
10^{-11}$ m2/s, and $\Pi(\varphi)=0.13$–80 kPa for $\varphi=0.05$–0.34 (see
Figs. 3 and 4).

Equations (1), (2) and (5) have been solved numerically for an initial
concentration $\varphi_{0}=0.05$, $\Pi_{i}=50$ kPa, and various Pe. For
$\text{Pe}\ll\mathcal{O}(1)$, diffusion homogenizes the concentration during
the shrinkage of the bag as illustrated in Fig. 1(b). In this regime,
$\varphi(r,t)\simeq\varphi_{0}R_{0}^{2}/R^{2}$ owing to mass conservation, and
Eq. (1) shows that $\tau_{\text{eq}}\propto R_{0}/(\mathcal{L}_{p}\Pi_{0})$
[see Fig. 1(e)]. For $\text{Pe}\gg\mathcal{O}(1)$, Fig. 1(c) shows that the
concentration close to the membrane quickly tends to $\varphi^{\star}$ defined
by $\Pi(\varphi^{\star})\simeq\Pi_{i}$. The dynamics is then limited by the
relaxation by diffusion of the concentration gradient over the bag, and
$\tau_{\text{eq}}\propto R_{0}^{2}/D_{0}$ (see Fig. 1(e) and SM [27]). Figure
1(e) shows that, for a permeability of $\mathcal{L}_{p}=2\times 10^{-12}$
m/(Pa s) typical of dialysis bags used for osmotic compression,
$\tau_{\text{eq}}\simeq 1$ day for $R_{0}\simeq 1~{}$cm. These results also
show that increasing the permeability $\mathcal{L}_{p}$ does not decrease
equilibration times in this regime because the transport is diffusion-limited,
and this would only reinforce the importance of the concentration gradient
($\text{Pe}\propto\mathcal{L}_{p}$). Figure 1(e) indicates, however, that
reducing the bag size to the microfluidic scale ($R_{0}\leq 100~{}\mu$m)
should significantly accelerate osmotic compression.

To measure EOSs in short times, we thus developed a microfluidic chip
mimicking membrane osmometry at the nanoliter scale to improve mass transport,
and with a reservoir to impose a salt chemical potential. First, we made
poly(ethylene glycol) diacrylate (PEGDA) chips using a rapid prototyping
technique [37] (see SM for details [27]). Figure 2(a) shows their design: two
parallel channels (AB and CD) are connected by a transverse channel of height
$h=45~{}\mu$m, width $w=100~{}\mu$m, and length $L=650~{}\mu$m. To integrate a
nanoporous membrane, we then inject an aqueous formulation containing PEGDA, a
photoinitiator, and poly(ethylene glycol) (PEG) and use spatially-resolved
photo-polymerization [38] to cross-link a hydrogel at one end of the
transverse channel. PEG chains induce a phase separation during polymerization
leading to a nanoporous hydrogel [39]. After polymerization, the hydrogel is
flushed leading to the nanoporous membrane of width $\simeq 25~{}\mu$m shown
in Fig. 2(b). Such membranes withstand pressure drops up to several bars
thanks to the chemical bonds with the acrylate groups of the channel walls,
and they reject nanoparticles $\simeq 20$ nm in diameter while being permeable
to water and salts [38].

Figure 2: (a) Design of the microfluidic chip. (b) Bright field view showing
the nanoporous membrane in the transverse channel. (c) Compression of the
dispersion by an imposed pressure, $P_{i}$, and (d) typical fluorescence
image. (e) Space-time plot of $\varphi(x,t)$ for $P_{i}=19~{}$kPa and
$c_{\text{res}}=1$ mM. (f) Continuous acquisition at $c_{\text{res}}=5$ mM
using a pressure ramp after a constant pressure step. The imposed pressures
$P_{i}$ vs $t$ are shown with white dotted lines on the space-time plots.

For EOS measurements, we studied dispersions of polystyrene particles
stabilized by sulfate charges to avoid charge regulation effects
(ThermoFisher, S37200, Lot No. 1677598). According to the manufacturer,
$2a\simeq~{}26\pm 3$ nm and $Z\simeq 450$ from titration. To perform
measurements of concentration profiles, this dispersion is seeded with a tiny
amount (concentration ratio $\simeq 2.7\times 10^{-4}$) of fluorescent
colloids of the same size and the same chemistry (ThermoFisher, F8845, Lot No.
2059222). We checked that the fluorescence intensity varies linearly with the
colloid concentration, and that photobleaching is negligible thanks to a
shutter limiting the overall exposure.

In a typical experiment, we first impose $P_{C}-P_{D}\simeq 1$ kPa to ensure a
steady flow ($\simeq 30~{}\mu$L/h) of an aqueous sodium chloride (NaCl)
solution at concentration $c_{\text{res}}$ ranging from $0.1$ to 100 mM. Then,
we fill channel AB with the dispersion at a volume fraction of $\simeq 0.075$.
Colloids invade the transverse channel by diffusion and diffusio-phoresis
induced by any ion concentration gradient, as shown in a similar geometry
[40]. After enough colloids have invaded the transverse channel, we flow an
inert oil (AR200, Sigma-Aldrich) in channel AB to trap the colloids upstream
of the membrane in the volume $hwL\simeq 3$ nL [see Fig. 2(c)]. In a typical
experiment, we impose a fixed pressure $P_{A}=P_{B}$ at inlets A and B, and
acquire fluorescence images [see Fig. 2(d) for a typical image]. The pressure
drop across the transverse channel, $P_{i}=P_{A}-(P_{C}+P_{D})/2$, ranges from
$10$ to 100 kPa, and is higher than the capillary pressure drop ($\leq 1$ kPa)
across the oil/dispersion interface.

Figure 2(e) shows a space-time plot of the concentration profile
$\varphi(x,t)$ estimated from the fluorescence images averaged over the width
$w$ for $P_{i}=19~{}$kPa and $c_{\text{res}}=1$ mM. The pressure drop
compresses the dispersion upstream of the membrane, therefore concentrating
the particles. After a transient $\simeq 300$ s, a steady state is reached
with the homogeneous concentration $\varphi^{\star}\simeq 0.16$. In such
experiments, the thermodynamic equilibrium corresponds to the balance between
the imposed mechanical pressure and the osmotic pressure of the dispersion,
$\Pi(\varphi^{\star})=P_{i}$, and thus to one point of the EOS, measured here
in a few minutes.

To get the entire EOS, $P_{i}$ is increased up to 100 kPa by steps of 10 kPa,
and the concentration upstream of the membrane is measured when equilibrium is
reached. Figure 3 shows such data for different reservoirs conditions. The
measurement accuracy allows one to highlight the screening reduction due to a
salinity decrease (increase of $\Pi$ at fixed $\varphi$ for decreasing
$c_{\text{res}}$), even when approaching the salt-free limit.

Figure 3: Microfluidic acquisition of EOS. Open symbols indicate colloids
aggregated irreversibly by the compression. The green dots were obtained from
the continuous acquisition shown in Fig. 2(f). The plus signs correspond to
the out-of-equilibrium measurements using Eq. (7), see text. Salt
concentrations are $c_{\text{res}}=0.1$ (blue), $1$ (orange), $5$ (green),
$10$ (red), and $100$ (purple) mM. The colored lines were obtained from
solutions of the OZ equation. The black line was obtained from PFBD
simulations at $c_{\text{res}}=0.1$ mM. Snapshots are volume renderings of the
electric potential field in this simulation at three concentrations.

After each point acquisition, $P_{i}$ is set to $0$ in order to check the
redispersibility of the particles upstream of the membrane (see Fig. 3). The
particles aggregate regardless of the imposed pressure for
$c_{\text{res}}=100$ mM, but remain redispersible up to $\Pi=100~{}$kPa for
$c_{\text{res}}\leq 1~{}$mM. This point by point acquisition was also compared
to a continuous acquisition in which $P_{i}$ has been slowly increased up to
$P_{i}=100$ kPa during $\simeq 6$ h after an initial step at $P_{i}=5.5$ kPa.
For this experiment at $c_{\text{res}}=5$ mM, the slow pressure ramp ensures
quasi-equilibrium with almost homogeneous concentration [see Fig. 2(f)]. As
shown in Fig. 3, the corresponding $\Pi(\varphi)$ curve perfectly overlaps the
point-by-point acquisition, demonstrating the possibility to determine EOS
with an unprecedented resolution.

These experimental EOS have not been compared to measurements made by
classical osmotic compression as this would require too large volumes of the
commercial dispersion, and would therefore be prohibitively expensive.
Instead, they are compared to the theoretical estimates obtained from the
solution of the OZ equation with renormalization in Fig. 3. Note that this
same theoretical framework has already been successfully used to describe EOS
measured by classical osmotic compression for a dispersion very close to the
system studied here [41]. For the dispersion considered here, perfect
agreement over the full range of $\varphi$ and $c_{\text{res}}$ is obtained
using the measured bare charge $Z=450$, and a unique colloid radius $a=10$ nm
slightly smaller than the value measured by dynamic light scattering, $a\simeq
11$–$13$ nm with a polydispersity index PdI$=0.1$–0.2 in line with other
measurements on the same dispersion, evidencing also a broad size distribution
[42]. We believe this small mismatch is primarily due to polydispersity, while
calculations strictly consider monodisperse colloids.

Although there is no universal model or scaling behavior for the osmotic
pressure in the range of volume fraction and salt concentration considered
here, a few comments on the physics underlying the EOS shown in Fig. 3 are in
order. The compressibility factor $Z=\Pi/(nk_{B}T)$ (with $a=10~{}$nm)
reported in Fig. S2 in the SM [27] is always much larger than unity (typically
in the range 10–200) because of the significant electrostatic repulsions at
the finite volume fractions studied here. Long-range interactions come into
play fairly quickly for all but the highest salt concentration. Therefore, an
”effectively concentrated” regime in which the mean intercolloid distance is
less than a few times the interaction range is obtained for $\varphi$ greater
than a few percents. Under these conditions, colloids are trapped in
electrostatic cages, leading to a solidlike (glassy or crystalline) state
whose thermodynamic properties are well described by the simple, albeit
nonlinear, Poisson-Boltzmann cell model which assumes that the osmotic
pressure of the dispersion is dominated by the contribution of ions [43, 44,
45, 46, 47, 41]. This model, used with the same physicochemical parameters as
those used for the OZ theory, indeed allows one to describe most of the
experimental points quite well (see Fig. S2 in SM [27]). However, the high
colloidal surface charge and associated significant ion condensation prevent
any analytical scaling law from being obtained. Data points at intermediate
salt content and volume fraction which are not well described by the cell
model correspond to a liquidlike phase in which colloids interact at a non-
vanishing distance (compared to their size). In this regime, the osmotic
pressure is dominated by the contribution of direct colloid-colloid
interactions and can be estimated from perturbation methods or solutions of
the OZ equation (see Ref. [41] for more detailed discussions on the different
methods to estimate EOS of charge-stabilized dispersions).

As the use of effective potentials and closures with the OZ equation remains
an active research topic especially under concentrated and low salt conditions
[15], we also performed particle scale numerical calculations using Brownian
dynamics coupled to a three-dimensional nonlinear Poisson-Boltzmann solver
providing the many-body electrostatic forces exerted on each colloid
(Particle-Field Brownian Dynamics, PFBD, see SM [27]) [48, 49, 47, 50, 51].
This method is more precise as it does not rely on approximate closure
relations and does not require the definition of an effective potential. As in
the experiments where compression occurs in a single direction, compression is
driven along a given axis in the simulation. Because these numerical
calculations are long, they have been undertaken only for $c_{\text{res}}=0.1$
mM, the most difficult case to capture with the renormalized OZ approach. The
close agreement among simulation, theory, and experiments [see Fig. 3]
confirms again the robustness of the EPC renormalization used with liquid
theories, and also shows that the anisotropy of the compression, as compared
to the isotropic case considered with OZ theory, does not play a role in our
configuration.

We now exploit the time-resolved concentration profiles during the compression
to measure the collective diffusion coefficient $D(\varphi)$. In the framework
of the local thermodynamic equilibrium, the imposed pressure drop $P_{i}$
induces a mean flow in the channel given by:

$\displaystyle J=\mathcal{L}_{p}\\{\Pi[\varphi(x=0,t)]-P_{i}\\}\,,$ (7)

because the pressure drop due to the viscous forces is negligible. In this
one-dimensional model, the colloid volume fraction obeys:

$\displaystyle\frac{\partial\varphi}{\partial
t}+J\frac{\partial\varphi}{\partial x}=\frac{\partial}{\partial
x}\left(D(\varphi)\frac{\partial\varphi}{\partial x}\right),$ (8)
$\displaystyle J\varphi(x=0,t)=\left(D(\varphi)\frac{\partial\varphi}{\partial
x}\right)_{x=0,\,t},$ (9)

with Eq. (9) ensuring the rejection of the colloids by the membrane. Again in
this model, $\Pi(\varphi)$ and $D(\varphi)$ are assumed in Donnan equilibrium
with the salt concentration $c_{\text{res}}$ imposed upstream of the membrane.

We first exploited $\varphi(x=0,t)$ vs $t$ measured during the first steps of
imposed $P_{i}$ [see Fig. 2(e)], to estimate the EOS using Eq. (7) and known
values of the permeability $\mathcal{L}_{p}$. $J\simeq\dot{A}/w$ is measured
from the temporal evolution of the area $A$ of the dispersion estimated from
the fluorescence images [see Fig. 2(d)]. The data $\Pi[\varphi(x=0,t)]$ vs
$\varphi(x=0,t)$ measured this way are shown in Fig. 3 with
$\mathcal{L}_{p}\simeq 1.7\pm 0.3\times 10^{-10}$ m/(Pa s) in agreement with
values measured using the same technique and only water upstream of the
membrane ($\Pi=0$). These out-of-equilibrium data at low $\varphi$ are in good
agreement with the theoretical EOS at equilibrium thus confirming the
hypothesis of local Donnan equilibrium, despite the transmembrane flow.

In a second step, we measured $\psi(x,t)=\int_{0}^{x}\varphi(u,t)\text{d}u$,
the amount of colloids trapped upstream of the membrane up to the position
$x$. Temporal variations of $\psi(x,t)$ are related to the colloid flux at
position $x$ [see Eqs. (8) and (9)] and can thus be used to estimate
$D(\varphi)$ providing we compute $\partial_{x}\varphi$ and $J$ from the data.
The inset of Fig. 4 shows such analysis for various $x$ at $c_{\text{res}}=1$
mM and 10 mM.

Figure 4:  $\varphi(x,t)$ at various $t$ for $P_{i}=6~{}$kPa and
$c_{\text{res}}=10~{}$mM. Black lines are numerical solutions of Eqs. (7), (8)
and (9) with $\Pi(\varphi)$ and $D(\varphi)$ calculated from liquid theories.
Inset: Theoretical $D(\varphi)/D_{0}$ at various $c_{\text{res}}$, same color
code as in Fig. 3. Circles are measurements at $c_{\text{res}}=1$ and 10 mM.

As expected, $D(\varphi)$ is significantly larger than the Stokes-Einstein
$D_{0}$ due to the electrostatic repulsion, in particular when the dispersion
is in equilibrium with a low salinity reservoir. These data are compared in
the inset of Fig. 4 to the theoretical estimates given by the methods
explained above, showing a very good agreement. To further confirm our model,
we numerically solved Eqs. (7), (8), and (9) with $\Pi(\varphi)$ and
$D(\varphi)$ estimated theoretically for $c_{\text{res}}=10$ mM, using the
experimental initial condition $\varphi(x,t=0)$ at $P_{i}=6$ kPa. Figure 4
shows that the overall dynamics is well predicted by the model, without any
free parameters, confirming the assumption of Donnan equilibrium despite the
fluid flow.

In the above comparison, the theoretical $D(\varphi)$ is estimated using the
short-time sedimentation hindrance function $K_{s}(\varphi)$, showing that
$K_{s}\simeq K$ for these charged colloids up to $\varphi\simeq 0.15$. Such a
similarity has already been shown numerically for hard-sphere dispersions
[52], but experimental evidence is still lacking for polydisperse and
concentrated dispersions. The range of experimental parameters explored above
leads to unambiguous measurements of the long-time $D(\varphi)$ only in the
range $\varphi\simeq[0.05$–0.15$]$ and new experiments are now needed to
access a wider $\varphi$ range, especially to investigate the possible role of
many-body hydrodynamics.

In this work, we developed a chip to measure with improved resolution and
acquisition time, EOS of charged dispersions in equilibrium with a salt
reservoir down to osmotic pressures of a few kPa. The dynamics can also be
used to measure simultaneously the long-time collective diffusion coefficient
of the dispersion, also in Donnan equilibrium, a quantity difficult to measure
using classical means. In the future, we plan to use these tools to address
the breakdown of the Donnan equilibrium hypothesis, a subtle issue that could
be relevant to many processes, such as ultrafiltration, that involve
potentially intense flows.

###### Acknowledgements.

We thank M. Guirardel, H. Fay, J. Castaing, P. Bacchin and M. Meireles for
fruitful discussions. This work benefited from meetings within the French
working group GDR CNRS 2019 Solliciter LA Matière Molle (SLAMM). We also
acknowledge Solvay and the ANR program Grant No. ANR-18-CE06-0021 for
financial support. This work was performed using HPC resources from GENCI-
CINES/IDRIS (Grant No. A0080911444) and CALMIP (Grant No. 21BP21039R1).

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