Transient separation of 2 chemicals in 1D, due to their different diffusion rates¶

Starting with identical concentrations in one of the bins, they initially separate...¶

until they eventually attain identical concentrations at equilibrium¶

For a complete separation by means of membranes, see experiment membranes_3

TAGS : "diffusion 1D", "basic"¶

In [1]:

LAST_REVISED = "Aug. 15, 2025"
LIFE123_VERSION = "1.0.0rc6"        # Library version this experiment is based on

In [2]:

#import set_path                    # Using MyBinder?  Uncomment this before running the next cell!

In [3]:

#import sys
#sys.path.append("C:/some_path/my_env_or_install")   # CHANGE to the folder containing your venv or libraries installation!
# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try:  import set_path   

from life123 import BioSim1D, ChemData, PlotlyHelper, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Prepare the initial system¶

with identical bin concentrations of the chemicals A and B, in the center bin.
B diffuses much faster than A

In [5]:

chem_data = ChemData(names=["A", "B"], diffusion_rates=[0.1, 2.], plot_colors=["turquoise", "green"])

bio = BioSim1D(n_bins=9, chem_data=chem_data)

In [6]:

bio.set_bin_conc(bin_address=4, chem_label="A", conc=10.)
bio.set_bin_conc(bin_address=4, chem_label="B", conc=10.)

bio.describe_state()

SYSTEM STATE at Time t = 0:
9 bins and 2 chemical species

Out[6]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.0	0.0	0.0	0.0	10.0	0.0	0.0	0.0	0.0
1	B	2.0	0.0	0.0	0.0	0.0	10.0	0.0	0.0	0.0	0.0

In [7]:

bio.system_snapshot()

Out[7]:

	A	B
0	0.0	0.0
1	0.0	0.0
2	0.0	0.0
3	0.0	0.0
4	10.0	10.0
5	0.0	0.0
6	0.0	0.0
7	0.0	0.0
8	0.0	0.0

In [8]:

bio.visualize_system(title_prefix="Diffusion.  The 2 concentrations initially perfectly overlap")   # Line curve view

In [9]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Request history-keeping for some bins¶

In [10]:

# Request to save the concentration history at the bin with the initial concentration injection, 
# and at a couple of other bins
bio.enable_history(bins=[0, 2, 4], frequency=3, take_snapshot=True)    

History enabled for bins [0, 2, 4], for ALL chemicals

In [ ]:

Initial Diffusion Step¶

In [11]:

# Advancing to time t=1, with moderate time steps
bio.diffuse(total_duration=1.0, fraction_max_step=0.4)

Out[11]:

{'steps': 16, 'system time': '1.0667', 'time_step': 0.066666}

In [13]:

bio.describe_state()

SYSTEM STATE at Time t = 1.066656:
9 bins and 2 chemical species

Out[13]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.000031	0.001396	0.044318	0.876317	8.155875	0.876317	0.044318	0.001396	0.000031
1	B	2.0	0.397538	0.694625	1.209567	1.725287	1.945964	1.725287	1.209567	0.694625	0.397538

In [ ]:

The 2 chemicals `A` and `B`, previously identical in concentrations, have separated due to their different diffusion rates¶

Notice how B has spread out far more than A

In [14]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [15]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Let's advance the diffusion¶

In [16]:

bio.diffuse(total_duration=2.0, fraction_max_step=0.4)

Out[16]:

{'steps': 31, 'system time': '3.1333', 'time_step': 0.066666}

In [17]:

bio.describe_state()

SYSTEM STATE at Time t = 3.133302:
9 bins and 2 chemical species

Out[17]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.002127	0.027200	0.270603	1.770845	5.858450	1.770845	0.270603	0.027200	0.002127
1	B	2.0	1.009997	1.057293	1.129782	1.193555	1.218748	1.193555	1.129782	1.057293	1.009997

`B` is close to equilibrium, while `A` is nowhere near it!¶

In [18]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [19]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Let's advance the diffusion some more¶

In [21]:

bio.diffuse(total_duration=2.0, fraction_max_step=0.4)

Out[21]:

{'steps': 31, 'system time': '5.1999', 'time_step': 0.066666}

In [22]:

bio.describe_state()

SYSTEM STATE at Time t = 5.199948:
9 bins and 2 chemical species

Out[22]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	0.012045	0.087799	0.524250	2.107562	4.536688	2.107562	0.524250	0.087799	0.012045
1	B	2.0	1.097383	1.103807	1.113648	1.122302	1.125720	1.122302	1.113648	1.103807	1.097383

Notice that the concentrations of `A` and `B` in the central bin 4 are again getting closer in value¶

In [23]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [24]:

bio.system_heatmaps(title_prefix="Diffusion")

In [ ]:

Finally, let's advance the diffusion to equilibrium¶

In [25]:

bio.diffuse(total_duration=95.0, fraction_max_step=0.4)

Out[25]:

{'steps': 1426, 'system time': '100.27', 'time_step': 0.066666}

In [26]:

bio.describe_state()

SYSTEM STATE at Time t = 100.26566:
9 bins and 2 chemical species

Out[26]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7	Bin 8
0	A	0.1	1.092097	1.100994	1.114625	1.126612	1.131346	1.126612	1.114625	1.100994	1.092097
1	B	2.0	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111	1.111111

All bins now have essentially uniform concentration¶

Notice that the concentrations of `A` and `B` in the central bin 4 are again essentially identical; the separation of `A` and `B` is coming to an end¶

In [27]:

bio.visualize_system(title_prefix="Diffusion")   # Line curve view

In [28]:

bio.system_heatmaps(title_prefix="Diffusion")

Verify that the total amount of each chemicals hasn't changed¶

In [29]:

# Let's verify mass conservation
bio.check_mass_conservation(chem_label="A", expected=10.)

Out[29]:

True

In [30]:

bio.check_mass_conservation(chem_label="B", expected=10.)

Out[30]:

True

In [ ]:

Visualization of time changes at particular bins¶

Instead of visualizing the entire system at a moment in time, like in the previous diagrams, let's now look at the time evolution of the concentrations of `A` and `B` at selected bins (bins whose history-keeping we requested prior to running the simulation)¶

In [31]:

bin4_hist = bio.conc_history.bin_history(bin_address=4)   # The bin where the initial concentration of `A` and `B` was applied
bin4_hist

Out[31]:

	SYSTEM TIME	A	B
0	0.000000	10.000000	10.000000
1	0.199998	9.607945	4.725956
2	0.399996	9.238843	3.251818
3	0.599994	8.891203	2.617687
4	0.799992	8.563635	2.254197
...	...	...	...
496	99.399006	1.132184	1.111111
497	99.599004	1.131988	1.111111
498	99.799002	1.131793	1.111111
499	99.999000	1.131600	1.111111
500	100.198998	1.131409	1.111111

501 rows × 3 columns

In [32]:

bio.plot_history_single_bin(bin_address=4, title_prefix="`A` and `B` separate, until they finally rejoin")

The transient separation of `A` and `B` is best seen by plotting the difference of their concentrations, as a function of time¶

In [33]:

bin4_hist["A"] - bin4_hist["B"]

Out[33]:

0      0.000000
1      4.881988
2      5.987024
3      6.273516
4      6.309437
         ...   
496    0.021073
497    0.020877
498    0.020682
499    0.020489
500    0.020298
Length: 501, dtype: float64

In [34]:

PlotlyHelper.plot_curves(x=bin4_hist['SYSTEM TIME'], y = bin4_hist['A'] - bin4_hist['B'], 
                         x_label="SYSTEM TIME", y_label="[A] - [B]", 
                         colors="purple", title="Transient separation of `A` and `B`, as seen in central bin 4")

In [ ]:

The transient separation of `A` and `B` is also seen in other bins...¶

In [35]:

bio.plot_history_single_bin(bin_address=0, title_prefix="The transient separation of `A` and `B` happens at other bins, too...")

In [36]:

bio.plot_history_single_bin(bin_address=2, title_prefix="The transient separation of `A` and `B` happens at other bins, too...")

In [ ]:

Transient separation of 2 chemicals in 1D, due to their different diffusion rates¶

Starting with identical concentrations in one of the bins, they initially separate...¶

until they eventually attain identical concentrations at equilibrium¶

TAGS : "diffusion 1D", "basic"¶

Prepare the initial system¶

Request history-keeping for some bins¶

Initial Diffusion Step¶

The 2 chemicals A and B, previously identical in concentrations, have separated due to their different diffusion rates¶

Let's advance the diffusion¶

B is close to equilibrium, while A is nowhere near it!¶

Let's advance the diffusion some more¶

Notice that the concentrations of A and B in the central bin 4 are again getting closer in value¶

Finally, let's advance the diffusion to equilibrium¶

All bins now have essentially uniform concentration¶

Notice that the concentrations of A and B in the central bin 4 are again essentially identical; the separation of A and B is coming to an end¶

Verify that the total amount of each chemicals hasn't changed¶

Visualization of time changes at particular bins¶

Instead of visualizing the entire system at a moment in time, like in the previous diagrams, let's now look at the time evolution of the concentrations of A and B at selected bins (bins whose history-keeping we requested prior to running the simulation)¶

The transient separation of A and B is best seen by plotting the difference of their concentrations, as a function of time¶

The transient separation of A and B is also seen in other bins...¶

The 2 chemicals `A` and `B`, previously identical in concentrations, have separated due to their different diffusion rates¶

`B` is close to equilibrium, while `A` is nowhere near it!¶

Notice that the concentrations of `A` and `B` in the central bin 4 are again getting closer in value¶

Notice that the concentrations of `A` and `B` in the central bin 4 are again essentially identical; the separation of `A` and `B` is coming to an end¶

Instead of visualizing the entire system at a moment in time, like in the previous diagrams, let's now look at the time evolution of the concentrations of `A` and `B` at selected bins (bins whose history-keeping we requested prior to running the simulation)¶

The transient separation of `A` and `B` is best seen by plotting the difference of their concentrations, as a function of time¶

The transient separation of `A` and `B` is also seen in other bins...¶